If your horse goes to canter out in the pasture or on the trail or something does he normally pick up his right or left lead first? If he's more comfortable on his left lead, then go to the right first. This way you will have only one turn on his right lead and two on his left. However, my horse likes his right lead better and I still run him to the right first because he just runs better in that direction. He has no problem picking up either of his leads. Whatever direction she runs best just stick with that in shows. I think that it's still a good idea to occasionally practice different directions though.
My mare and old gelding run to the right. I've never had to run a horse backwards (left) so hopefully my colt will go to the right. Most horses are left handed so they run better to the right.
How is there a backwards way when running to the right or left is acceptable? I would also suspect that like humans, some horses are righties and some are lefties. Where did you learn that most are lefties out of curiousity?