Isn't the ideal to have both eyes open when shooting?
Bamaboy is cross eye dominant, so was his grandad. We never got my Dad figured out 'till late in life........he missed a bunch of game.
Much like PetaW, I caught bamaboy squinting down the barrel of a toy gun with his "off" eye at about age 6. We swithched shoulders right then and now he shoots off his left shoulder with both eyes open. (pretty darn well I might add).
With a handgun, he shoots right handed, but when he presents the pistol, it is skewed slightly to his left, as he brings the gun into alignment with his dominanant (left) eye.
Learning from a young age, running the rifle from his "off" shoulder is not a big deal. I would not want to do that as an adult.