A few disjointed thoughts,
I've read books and articles that have said that corps dancers tend to be grouped into "rights" and "lefts" for ballet after ballet. It may have been in Toni Bentley's book Winter Season in that there was a statement that it's easier to learn a last-minute corps part to one's "natural" side.
In Merrill Ashley's memoir, she wrote about how Balanchine would choreograph or change to her stronger side most of the time. An example she gave in which he didn't was in Ballo Della Regina, in which he replaced pique turns, which didn't film well, for the Dance in America taping, to fouettes that were difficult for her, and then left them in the stage version.
Skaters in general spin and turn to one side only. It's a very big deal when a competitive skater does turns to both sides, and there's a special rule under the New Judging System for this, so that the double-sided spin is not counted as two individual spins. Ilia Klimkin is known for his camel spins to both sides. Jeff Buttle has done them, and I think Kwan had early in her career.
John Curry insisted that members of his skating companies be able to spin to both sides, although I don't think he required this of guest stars, like Dorothy Hamill.
It is a rare skater who spots jumps, because spotting slows down the rotation. Kevin van der Perren did spot, at least until last season. (I haven't seen any of his 2006 performances.) I'm guessing that the same thing is true of spins. It would be impossible to spot a layback spin.
Suzanne Farrell had a corner from which she made most of her Balanchine entrances. It would have looked odd if she had entered from the other side.
