If one shoots with both eyes open it does make a difference. The cast (angle of the butt stock to right or left) changes where the gun sits in front of your sight picture. Regardless of the action type and ejection to right or left, how the gun sits as you aim it makes a difference. As a lefty, my shotgun is cast so that the butt stock "bends" off to the right which then pushes the barrel and sight to the left and more to the front of my face. A neutral cast means tilting my head more to the left, gives a tilted sight picture and less of a natural hold to the gun. Over a round (25 shots) of trap, my lefty will give me on average a 2 hit better result than any of my side by side, over under or pump neutral cast guns. I have one gun with a distinct right hand cast and that's on average 1 hit lower still.