I think I know what you are talking about. While going to the right, my horse use to not get on the correct lead either. It took me about a half a year to a year for her to get on the correct lead. Try asking at a corner. Or trot in a small circle and while coming out of the circle, ask for a canter. And sometimes (if she is getting on the incorrect lead going to the right) then opening your reins a little and pull your horses head in (get your horse collected) and slightly pulling up on the inside rein and ask. If she gets on the wrong lead, stop imediatly and if she gets on the correct lead go around the arena (or wherever you ride) once or twice. Then slow to a walk and praise your horse! And just keep repeating this until she is always on the correct lead, going to the right. :)
Ask on a circle. Make sure she is bent to the inside, Sit the trot a few strides, Squeeze until she canters. It will be extremely hard for her to pick up the left lead when cantering a right circle if properly bent. If this still doesn't work, you may have to check saddle fit or some other medical problem. It could be causing her pain to be bent in that direction or pain from anything else when she picks up that lead. It probably is her stiff side but you will just have to work on it. Good luck.
I had to look up OTTB because I'm a blonde. I think it means of the track TB?
Cheater way! - What you could do for now is ... i'm goign to get yelled at! I'm not saying this is at all correct but it will help you get the correct lead for now....
When you go to ask for the canter, pull his head to the outside, it will force his inside leg forward and put him onto the correct lead, now it won't win you first place in a show, but it'll get you on the correct lead for now. You need to get him up underneath himself and balanced by doing tons of slow small circles and serpentines before you are going to be able to get him to pick it up properly.
What I do is if im going to the right, I ask my horse to canter while puttin my left foot back by the horses rump...and if im going to the left, I put my right foot back by the horses rump.
It helped me!! :]
If ur confused, just ask!!! Ill try to explain myself better!