Ok, this is an aspect that is related to throwing and catching the yo-yo, and for various kinds of loops. Understanding it may help to understand how a lot of different things work, so I thought it'd be a good topic to tackle.
Throughout, I'll refer to "sweet" and "sour". Mainly "sweet" is easier to do or more stable, while "sour" isn't. I do this so I can stay away from words like "right" and "wrong", as there is usually never a real right and wrong way...
First, do you remember one of the simplist tricks of all? Where you make the yo-yo go up and down while keeping your palm down?
Try it now, and notice how the string switches sides as it goes down and comes up. If it starts away from you, it'll be on the side nearest you once it reaches your hand again, and vice versa.
When a yo-yo is wound up, it can only unwind one way, and the way it winds up depends on what direction the yo-yo is spinning. If the string is farthest from you with the yo-yo wound up, and you let it fall, the top of the yo-yo moves toward you as it unwinds.
When it gets to the bottom of the string the yo-yo is still turning the same way (the top toward you). Now, when the yo-yo winds up, it keeps turning in that direction. The power given to the spin by gravity and your throw as it unwinds is used to wind it back up.
But now the yo-yo is wound up the opposite way from when you started. Now the string is on the side closest to you. If you let it drop, the yo-yo will spin so that the top moves away from you, and will keep that spin until it reaches your hand and stops again.
So, the first thing to keep in your mind is that the way the yo-yo is wound up, is what effects the spin the yo-yo has as it unwinds, and thus how it winds up.
Another thing to keep in mind is that it has the same spin now matter which dirction you throw it (staying on the same plane). If it is would up so that the string comes off the side farthest from you while your palm is down, then you can let it fall, throw it forward, upward, behind you, etc. and the top of the yo-yo will always be spinning the same way (in this case toward you).
Now instead of actually throwing it in all directions, you may just want to unroll it a little with your other hand to see that it unwinds the same way. If you actually try throwing and catching it in those different directions, some of them will be pretty hard or akward to do, and that is something I'll explain later.
Also, be careful on how you move your hand. The next important thing to know is that flipping the yo-yo over reverses how it unwinds. If the yo-yo is in your hand and your palm is down, if you turn your hand so it is palm up, everything has now been reversed. Try unrolling the yo-yo with your other hand and notice that it now turns the opposite way as it unrolls. This is used frequently in yo-yo tricks in order to make life easier, so much that you probably don't even think about it anymore.
When you start yoing, and get past up and down, you are usually told to have the yo-yo the "right" way in your hand, and then do a Throwdown using your arm.
When you do the old up and down where you keep your palm down and just move your hand up and down, it doesn't matter as much which way the yo-yo unwinds or rewinds. But at the same time, it is difficult to get any power into the downward motion. Using the Throwdown lets you get this power, but also introduces all of the Sweet and Sour stuff.
A good throwdown has your elbow bent so your hand is near your ear, and your wrist bent. Look at your hand as you throw, and see how it moves in an arc? If you think of that arc as a wheel turning, you'll see that it turns so that the top moves away from you.
Now, remember that the way a yo-yo is wound effects its directions of spin. The "right" way to have the yo-yo in your hand for a throwdown (palm up), is the string coming off the top of the yo-yo.
Now, unwind the yo-yo with your other hand, and see that the top moves away from you. So, the key thing is that the direction of spin of the unwinding yo-yo is the same as that of your hand as you throw it. It is in your hand the "sweet" way. This lets you throw the yo-yo easily and with a lot of power. Don't worry about how the catching catching works yet. I'll cover that in the next section.
So, can you have the yo-yo in your hand the other way? With the string coming off of the bottom? The answer is yes. But, remember yo-yo unwinda the same way no matter how you throw it unless you flip it over. Even though your hand is moving away at the top, the yo-yo still has to unwind with the top moving toward you. It is this conflict of the two moving in opposite directions that makes having the yo-yo in your hand the "sour" way more unstable.
The yo-yo "wants" to unwind the same way you're throwing it. The only way for it to do that is to be flipped around. So unless you throw the yo-yo very carefully (and a butterfly helps), then the yo-yo will likely turn sideways as you throw it and go out of control.
With a regular forward pass, the circular motion of the throw is the opposite direction, and so the yo-yo needs to be wound up the opposite way. Now, the lucky thing for us is again that flipping over the yo-yo changes the spin.
So, if the yo-yo is in your hand the "sweet" way for a throwdown, and you flip your hand over for a forward pass, the yo-yo is still the "sweet" way. And again, you can still do a forward pass with the yo-yo the other way, it is just less stable.
So, why would you want the yo-yo in your hand the "sour" way for either? Usually you wouldn't, but the direction of spin in the yo-yo affects things such as what string tricks are easier to do. If you wanted for instance to do a throwdown and then go into hydrogen bomb with the yo-yo being non-catchy, an easy way might be to put the yo-yo in your hand the "sour" way. There are also times when it might be convenient in looping. But most of the time you'll want to stay away from them due the unstability.
While I'm mentioning mainly the throwdown and forward pass, this all applies to almost any throw from breakaways to UFOs. The you turn the plane of the yo-yo, the rules just turn with it. If the yo-yos is in your hand the "sweet" way, and you turn your hand, it is still "sweet" since your hand and yo-yo turn together.
As long as the back of your hand is facing your target, it is still a normal throw (see the seperate section on palm-forward throws). Just remember that if you turn exactly 180 degrees from where you were, the yo-yo is still on the same plane as it was, just now everything is reversed, and the throw probably has a different name.
This document will just deal with things involving the yo-yo staying the same or turning 180 degrees at a time (both of which keep it in the same plane), so if you want to apply this to sideways or horizontal throws or anything, it should all apply exactly the same way.
One last thing is that if you look at a throwdown while the yo-yo is in your hand the "sweet" way, the way that it wants to unroll also lets it easily roll off of the tips of your fingers. If it is in your hand the other way, it rolls against your fingers. This doesn't make too too much of a difference in the standard throws because your hand is moving away from the yo-yo as you throw it, there tends not to be much friction. It makes a bigger difference in the palm-forward throws though.
This describes a specific kind of throw that isn't used that often, but is still interesting. Feel free to skip this if your worried about getting confused and just want to read about the regular things.
Now let's revisit the origional up and down one more time. In the regular throwdown and forward pass, you are pulling your hand away from the yo-yo and your hand is between the yo-yo and the target before you throw. One the other hand, in the up and down, you are pushing on the yo-yo more, and the yo-yo is between your hand and the target.
As I said origionally, it is harder to get power into the throw, but it can be done. You can bend your wrist a bit and try to get a more circular motion going to your throw. Also how the yo-yo rolls out against your hand makes more of a difference, since you're pushing on the yo-yo to start with.
Let's say we want to do a forward pass type motion with the palm up. The circular motion of the throw has the top coming toward us a bit. If the yo-yo is in our hand with the string coming off the top (would be "sweet" in the other kind of throw), as it unwinds, the top will be going away from us. This makes the spin part "sour". However, it roll off of your fingers easier, and this makes more of a difference in this kind of throw, making that part "sweet". And the reverse is true for the yo-yo in you hand the other way.
So what does this mean? If the string is coming off of the top, the fact that if can roll off of your fingers easier means you can put more power into the throw. But, the fact that the spin is different from your throw makes it easy to go out of control. If it is in your hand the other way, it is somewhat stable because of the spin, but it is difficult to get any power into the throw because of it being hard to roll off of your fingers.
If I had to choose one, I think I would go for the string on top, for several reasons. First it will probably be in your hand that way anyway since it is the "sweet" position for normal throws. Second, if you can't get any power in the throw, the stability is kind of moot, and if you have a butterfly, it isn't as much of an issue.
Why use this kind of throw at all? Mainly it looks different and maybe has some different possibilities. Besides the up and down, Kyle Nguyen has proven that you can use this kind of throw for string tricks, and I made a looping transition a while ago that used this to shove the yo-yo down from a moon quickly.
Ok, throwing is all fine and good, but what about catching stuff? In throwing there's a big issue about the throw spin being the same as the yo-yos spin. But in catching, the spin of the yo-yo has already been established, and you don't move your hand that much in the catch.
So, the main factors are how the spin of the yo-yo affects things as it hits your hand, and what your next throw is going to be.
Remember with the throw, when the yo-yo was in your hand the "sweet" way, as it unrolled, it also smoothly rolls off the tips of your fingers. If you now roll it back, you'll see that it rolls right back along your fingers into your palm. It also puts you in the right position for a "sweet:" throw. This is the "sweet" kind of catch. It is very comfortable and feels pretty natural.
If you have it the opposite way, when as it rolls up, it tends to want to roll toward the tip of your fingers and not toward your palm. It is also now in the position for a "sour" throw. So this is the "sour" catch. The fact that it rolls toward your fingers makes it harder to catch reliably especially if the yo-yo is spinning fast. It also tends to put more stress on the slipknot (catching the yo-yo the "sour" way can actually be used to break your slipknot on purpose).
Now, the tricky thing is remember that when you throw a yo-yo, it rolls up the opposite way as it started. So we now have the dilemma. Throw a forward pass the "sweet" way. Now try to catch it keeping your palm downward. You can feel how akward the catch is. Since the yo-yo has already left your hand, you can't change the spin on it (without getting into loops and things), so the solution is to reverse your hand instead.
This is why in a normal forward pass, you usually throw palm down, and catch palm up. This lets you have "sweet" throws and catches. The same thing for the throwdown, or UFOs, or anything.
As always, the same thing applies to the opposite. If you do a throwdown or punch type of throw and try to catch the yo-yo palm up, it is fairly difficult.
So, now we know why certain throws and catches are easier or harder, but what about some actual tricks?
The planet hop is probably the easiest example. Generally you start with a throwdown. Then have the yo-yo flip over your hand toward you, then away from, you etc. back and forth. This is the "sweet" kind of planet hop.
If you want to try the "sour" way, start with a throwdown, then hop away from you, then toward you, etc. You should notice that this is much less stable, even with a butterfly.
What is happening, is that each new hop is like a new throw without stopping for a catch. The same rule about the reversing of spin as it leaves your hand again is the same here.
As you do the throwdown, the spin is such that the top of the yo-yo moves away from you. Then it winds up keeping the same spin until it winds up all the way. Now the yo-yo is wound up the opposite way, and so when it unwinds again, the spin will have the top of the yo-yo moving toward you. And so for the spin of the yo-yo and motion of the "throw" to be the same, it is easiest to flip the yo-yo toward you the first time.
It may seem odd, but even though the yo-yo is in constant motion, and moves fluidy, the spin of the yo-yo is actually stopping and reversing every time it flips over your hand.
So, for the "sour" planet hop, again the spin of the yo-yo has the top moving away from you. As it winds up and unwinds, the spin is now moving toward you. If at this point you flip the yo-yo away from you, the motion of the yo-yo and the spin of the yo-yo are at odds with each other and this makes it unstable. The yo-yo "wants" to have these two be the same, so it'll try to have the yo-yo go sideways. Having a butterfly helps, but it is still less stable overall. However, flipping tricks actaully take advantage of this instability to create a new kind of stability (more on this in the flipping tricks section).
Now in a planet hop or reach for the moon, the yo-yo is constantly moving back and forth. If done the "sweet" way, this makes for a harmonious trick that is constantly "sweet". However, the other kind of non-flipping trick has the the yo-yo going over your hand in the same direction all the time.
For this I'll give the example of a non-flipping hop the fence. Pretty much what happens is that the trick alternates "sweet" and "sour" motions each time. If you start with a throwdown, the top of the yo-yo moves away from you. As it unwinds the second time it moves toward you. If at this point you jump the yo-yo over your hand away from you, the two are in conflict, and it is a "sour" motion. However, as it unwinds the third time, the top of the yo-yo moves away from you, and so if you hop the yo-yo over your hand away from you again, the two are in synch, and it is a "sweet" motion.
A lot of people when learning hop the fence, use a non-flipping technique, and so have this alternating "sweet" and "sour" motions, which gives an odd feel to the trick. It also means that the half of the time, the yo-yo is more likely to go out of control. Using a butterfly helps, but there is still this unstability there.
The other main kind of trick is those that have the yo-yo flip 180 degrees each time. For the yo-yo to move around in a circle the same direction (when the spin is constantly being reversed with winding and unwinding), the only two things to happen is to alternate between "sweet" and "sour" (as in the non-flipping hop the fence), or for the yo-yo to flip each time.
As an easy example, try throwing several forward passes in a row, catching it each time. Now analyze what you are doing. You throw the yo-yo palm down, as it returns you catch it palm up, and then you turn your palm downward again to throw it. Wait until the last second to turn your hand over to catch the yo-yo and immediately turn it over again to throw. You can see how you're grabbing the yo-yo and flipping it over each time. This is essentially what is going on with a inside loops, except you're using physics and technique to flip it without having to manually turn it over with your hand.
A flipping trick basically does not have "sweet" and "sour." Since it flips over for every loop, each one is a "sweet" loop. However, the direction the yo-yo is moving compared with the direction of the spin is what would be considered "sour" for a non-flipping trick.
As you release the forward pass, the spin of the yo-yo has the top moving toward you. The yo-yo winds up, and as it unwinds again, the top is now moving away. Now, the motion of the yo-yo in an inside loop is always toward you over the top, so now the motion of the throw and the spin of the yo-yo are in conflict. The yo-yo wants to flip over so they can be in harmony, but instead of fighting it as in a non-flipping trick, we go with the tendancy, making it flip over an entire 180 degrees.
The fact that the yo-yo is at an angle compared to the motion of the loop (usually 1 o'clock for right-handed or 11 o'clock for left-handed), and the technique that you develope are what lets this happen.
So, I hope this has been helpful. Some things to keep in mind. Butterflies are generally more stable for any non-flipping trick, especially if it is "sour". Flipping tricks are generally easier with an imperial/modified yo-yo due to the need for the yo-yo to flip over 180 degrees easily.
Just about any looping trick is some comination or variation on all of this. Something like horizontal loops is just flipping loops turned sideways. Outside loops is the same thing, but being controlled from a different part of your hand. Regular Stalls are a combination of a non-flipping planet hop type motion, combined with what is basically catching and throwing.
The reason the loop after a braintwister is easy even with a butterfly is that it is a "sweet" planet hop type of motion and not a flipping loop.
So, even if you aren't a "looping" type of player, this is important to know for transitions, since the current spin of the yo-yo determines what kind of planet hop or stall type motion you use regenerate in a "sweet" way. And if your freestyle is with a yo-yo that can handle some flipping tricks here and there, when it is best to use that kind of regneration (as with warp drive or triple trapeze).
Have fun... =)