I can think of two answers.
Field hockey and Frank Conner.
Bob
Bob, Frank Conner PLAYED IN the U.S. Open for tennis. He and Ellsworth Vines (I think, maybe it was Bill Tilden) are the only two that competed in the US Open for tennis and golf. There is a US Open champion in another sport that played the Senior Tour. Good guess, but no cigar.
I don't know anything about field hockey. I'd assume you can play L or R like regular hockey. No? If field hockey is a good answer then there are three sports. It is similar to one of them.
John Moore, not one of them. There are no restrictions on which side you use in soccer.