This is one of those times when Western riders and English riders are probably going to disagree, and even dressage & hunt seat riders may disagree. Dressage defines a canter is a 3 beat gait, which starts with the outside hind foot/leg for the first beat, then the diagonal pair of outside front foot/leg together with inside hind for the 2nd “strong” beat and the final 3rd beat is the inside front, which is why when the horse is on the correct lead, the inside shoulder always appears to be in front of the outside shoulder. Dressage theory is that the horse running freely without a rider will pick up the “correct” lead for the circle he plans to make, and can also do an impromptu flying lead change if he then decides to turn the other direction. Therefore, logic dictates that dressage riders will bend the horse to the inside, which tells the horse to prepare for circles to the left or right, depending on which direction the horse is going in the ring. The rider tells the horse which lead to pick up by asking with the outside foot, with the horse bent correctly to the inside. If a horse makes a tight circle on the wrong lead, he can easily be off balance, to the extent of tangling up his feet. Race horses, however, are trained on the track to canter/gallop in only one direction, so when they are retrained to be dressage mounts, teaching them to pick up the other lead can get “interesting”, because the concept of cantering on both leads while carrying a rider simply is not in the horse’s vocabulary. From a dressage standpoint, bending the horse to the outside tells him to prepare to canter on the lead to make balanced circles or tight turns in that direction. If the horse is going to canter counter-clockwise in a ring, the horse can only make balanced turns and circles to the left without running into the fence or being potentially badly off balance. If the horse is going clockwise in a ring, he can also only make circles to the inside, which would be right circles and turns, again, so he doesn’t run into the fence or get his feet tangled up. Which is why in dressage flying changes every stride are always done done down the center line, or on the long side, and counter canter work (cantering on the “wrong” lead for you non-dressage riders) is not done until horse & rider are already adept at the basics.
That having been said, I have frequently seen hunt seat riders bending the horse’s head to the outside to hold the horse on the outside track, instead of bending the horse correctly to the inside and using the inside leg to hold the horse on the outside track. The part of dressage that first caught my attention was that while my hunt seat teachers had often considered “do X because I’m telling you to do it that way” to be an acceptable explanation, good dressage instructors understand & explain why doing it in a certain way works better than the other way or ways. Every discipline tends to think they have all the answers, but bending the horse to the inside to help the horse pick up the correct lead, and remembering that the first beat is the outside hind, does, at least, have logic and an understanding of how the horse actually moves on its side. I don’t think one could do multiple flying changes every stride with the horse incorrectly bent to the wrong side.