One of the most controversial techniques in gymnastics and tumbling.
Most coaches, gymnasts, and cheerleaders are trained to rebound from round-off with arms up. There's a small few, in my view, that believe and train their students with arms ending at horizontal when doing the round-off.
I'm one of them. But it's difficult to get big groups to do it this way when you only see them once per week for one hour. Many of them do private lessons with other coaches that teach and reinforce the arms up and rebound.
The technique analysis:
If you watch any video of any tumbling by really skillful people, you might notice that they get their arms up as soon as possible after the round off into back handspring. But let's slow it down more. Get your mouse pointer, or if you're on mobile, tap on the play button and then pause it when the feet make contact with the ground after the round off (choose a pass where they do a back handspring out of it, not a whip or a tuck). Now, where are the arms when the feet make contact on the floor? Most of them will have their arms to horizontal, ready to whip them back to a back handspring. It's the only way to accelerate the back handspring. If you make contact with feet on the floor with arms at vertical, you wouldn't have anything to accelerate the back handspring. You would only have a "back-bend"- like action. It would slow way down.
I have seen this. It is rare and painful to watch. All the back handspring combinations were powered by the legs bending and pushing- the arms stayed straight up and down by the ears throughout the series. She had to go in a deep squat each time to make it. Very aesthetically unappealing.
Now here's a screen capture of one of the best gymnasts in the world, Simone Biles. This is her first round off for her first pass. Notice the arms at Horizontal before going into her back handspring.
This is why a few of us who think differently from the norm train round offs like this. It's just more conducive to progressive back ward tumbling skills.
Here's a video From Steve Butcher doing a clinic and emphasizing the "no rebound" drill and "falling backwards."
Here's the video of that image with Simone Biles: