PORT ST. LUCIE, Fla.—The first time Mets outfielder Collin Cowgill remembers picking up a bat, he swung it from the right side of the plate. On the surface, this doesn't seem all that unusual: Cowgill writes with his right hand. He eats with his right hand. If he wanted to throw a punch, he would do so with his right hand.

But for reasons he doesn't quite understand, Cowgill throws with his left hand—a quirk that puts him in one of the strangest and most exclusive clubs in baseball.

...