There's nothing wrong with mounting and dismounting from both sides. Whenever I start or train horses I always make sure they're just as good to do the same thing on both sides (When first getting on their back, I make sure I work on both sides putting pressure and both sides putting weight in the stirrups, just like when you desensitize a horse with a plastic bag or flashy red flag. You don't just teach them not to spook on one side, you make sure they're good all around.), cause like you said you never know if something will come up and you'll need to. I'd rather be able to dismount on the right side in an emergency than to get off on the left and end up severely hurt or worse because it was "improper". ;) It is something that's taught to do in lessons or for show (I'm not sure as to why, my guess is it could be for unison or something to do with way back when when you had a sword to carry with you xP Like I said, not completely sure why). But over all, if you're really into things and wish to mount and dismount on one side only, your choice. But it's definitely not going to "mess up your horses training" by far. XP Like I said, safety first, hence why I like one side of my horse to be just as useful as the next.