Thanks for the input! I also have never particularly stressed over the hand position unless something weird was going on with it that affected the cartwheel. Often I would teach it with fingers facing the same direction, what ryantroop mentioned in his first method. And I certainly plan on going along with HC, I was just curious how others teach them to see if I have been doing them incorrectly all along or have been teaching them some wacky way up until this point.
When I do a cartwheel, my hands are very slightly turned in and I have never had a problem with cartwheels, round offs, or any form of back tumbling out of a round off, so it has obviously worked out okay for me. The HC at this gym wants fingers all the way turned in so they are "looking" at each other. It feels really awkward to me, probably because I have never done it that way, and I was curious about the reasoning behind it. I don't doubt the technique, I've just never seen it taught that way before.