I had an old (Well, not really old....about 10) gelding who had that same problem. He was a reiner, so I didn't understand why it was a problem. He would lunge fine, run aorund on his left lead, but th emoment you were on his back there was a problem. He had trouble spinning on the left pivot foot, too. We thought he was just being stubborn, becaue the gal we took him from worked consistently ont he right w/o any left work.
We later found out he had a left stifle injury that was causing him incredible pain on the left lead of the left pivot. However, it was nearly invisible to normal vet checks without the x-ray. The vet showed us the picture and it looked like a bone splinter sticking into him from the inside. The surgery would cost a very hefty amount to remove, so instead we retired him to a guided trail ride service who I informed full well that he is not to do anything on his left side.
Just a thought.
Pssh.I didn't pick up the wrong lead
It's called a counter canter...
...A very advanced maneuver.