Ok, so here are my thoughts. This is going to be a bit more rambling than my earlier technique topics, because it's covering a wider range of skills, and has a few elements that I haven't entirely figured out how best to coach myself.
Until you get to twisting double saltos, a roundoff is the most obscenely complicated and arguably the most difficult skill to learn on floor. The reason we teach it to beginners is not that it's easy enough for them to get, but that it's absolutely necessary for any higher level skills.
I don't think there's much debate about what a roundoff should look like; long hurdle, long step comming out of the hurdle (this often isn't emphasized enough by coaches), a late but fast turn of the upper body so that one hand ends up facing back the direction you came from by the time the hands hit the floor, legs come together as soon as the push off the floor is complete, and then the feet are pulled down and under with an agressive snap to hollow. It should be landed in a hollow, on toe, leaning backwards. The backwards lean on landing is necessary to set up for a good backhandspring; obviously this must be altered if the roundoff is to be followed by something other than a backhandspring.
This is far less complicated than a roundoff; all you do is snap to a tight arch, and when your hands hit, snap to a tight hollow. There really isn't much more to it.
The only really complicated part is the block off the hands and the snap down at the end, as this section must be altered depending on what skill comes next. If it will be followed by another handspring or a whip, the feet should be snapped under the gymnast, like a roundoff. If it will be followed by a salto, the feet should be snapped down behind the gymnast to deflect their power upward.
I'm still working out the details of how this last part can best be accomplished. It seems to me that there are two ways of getting this deliberate underrotation in the backhandspring. The first is to not snap down so hard. This extends the backhandspring and causes it to underrotate, setting up for a powerful upward punch. I hesitate to coach this, as it seems counterproductive to deliberately decrease your power. The other method is to bend the arms; the decrease in upward push off the hands causes the feet to come to the floor faster, and thus the backhandspring is underrotated. I hesitate to teach this one as well, though it seems to happen naturally as gymnasts develope more powerful roundoffs. As I haven't figured out my preferred method of coaching this myself, I neither encourage nor discourage either of the above techniques.
While we're on the subject of backhandsprings, does anybody have any good drills or tricks to fix problems usually encountered by kids with hyperflexible backs? You know the type; they underrotate their roundoffs and undercut the first half of their backhandspring, but because they can arch their backs so far, the backhandspring still feels smooth and fast to them, so they have trouble correcting it.
Anyway, on to other elements.
Comming out of the backhandspring, the chest and shoulders are openned to hit an (ideally) straight body position on contact with the floor. After the feet leave the floor, the gymnast pulls the knees up to the chest for rotation. I tell my kids that their head should be the last thing to flip, as the most common problem is throwing the head and chest back on takeoff. As the gymnast developes more powerful tumbling and a better set, he/she can add a slight delay before pulling into the tuck and a kickout at the end.
Chest and shoulders should be slightly more open on the set than for a tuck. After the set, there's not much to it except to stay tight until you see the ground. Ideally, the body should be completely straight, but it often ends up being done with a slight arch or hollow. In my experience, most gymnasts tend to fix this automatically when they start twisting.
Same as a tuck, only more so. What more is there to say?
I want to hear somebody else's input on this; I've been trying, without success, to learn it for about a year. I've never coached anybody at a level anywhere near high enough to work this skill.