I don't think there's a rule... If their mane falls to the left, it falls to the left, and if you try and make it go to the right, it will stick up along his neck. Sure, you can train a mane, but that's more to keep it flat, not change which side it falls on.
I googled dressage images and most of them were on the right but a few were on the left. I agree if the main is totally falling to the left really hard to make it go the other way sometimes, I guess it doens't matter, I always thought right was traditional, anyone else?
The mane is supposed to lay on the right for eventing. But it doesn't REALLY matter. Both of my eventers have their manes on the left ( it's just easier to let it go it's way when it's a shire double mane! I'm just thankful it's on one side!) and I have never had anyone tell me that I need to change it.
Some eventers, you can't tell because they have the horse's mane so short that it stick straight up. Makes it easier to braid.
When you get to the upper levels, most of the professionals have their horse's manes on the traditional right side.