I used to obsess about this. Doing some hunters (but mostly eventing, and little straight dressage), it just felt “wrong” to have the mane on the left. In the case of my retired mare, I would try to “train” the mane to the right - though gave up when it didn’t “stick” - but would wet the mane and braid it over to the right before shows, clinics, etc. where I wasn’t going to braid, but where I wanted a neat, tidy mane that lay flat. ON THE CORRECT SIDE :lol: I would remove the braids the morning of, wet the mane and comb it down, and it usually stayed there. For a day or two.
When I did braid her mane (recognized HTs, and for her inspection with her filly) it was always to the right; I also pulled and trimmed it on the right. Luckily she has a reasonably fine mane and allows me to pull it - despite the fact that she’s a sensitive redhead.
Her daughter - now 5 - has the same mane on the left thing going on (when her mane started growing in, I was hopeful! that she would “avoid the curse.” but…no. :sigh:) I took her to her foal inspection when she was 2 1/2 months old, and braided the wispy mane to the right. Heehee! That will probably be the last time she’s braided, since my goals with her are modest: local, unrecognized shows and eventing to BN, where I don’t feel the need to braid 
She is a Diva and DO NOT PULL MY MANE, WTH are you DOING!!! :eek:, so I use a Solo comb or scissor the mane creatively, keep it short, and at this stage of the game am not going to sweat which side it falls on. This is what happens when you finally grow old enough not to care what people whisper about you behind your back…