Are there left seat qualifications specific to actual handling of the aircraft?
No, it is a non-factor, although in the Navy there are aircraft you only conduct carrier landings from in the left seat. To be qualified as an instructor in that aircraft, you have to demonstrate the ability to fly the same pattern from the right seat (but not actually land on the carrier).

I'll tell a story about switching hands. I was an LSO (landing signal officer) for E-2s, which had yokes and a central throttle quadrant. You always flew the carrier approaches from the left seat, though. 

We had a pilot who had developed bad habits in the landing pattern. To break him of it, I had him fly the landing pattern from the right seat at the field, which just plain felt awkward since you never did it at the boat. This made him break out of the auto-flying and have to think his way around the pattern instead, and he was able to stop what he was doing wrong. He carried the fix back over to the left seat.