I did not show the right side at all, so I am not sure how you can make that statement. The point is that a+b is symmetric in a <-> b and a-b is anti-symmetric

I did not show the right side at all, so I am not sure how you can make that statement. The point is that a+b is symmetric in a <-> b and a-b is anti-symmetric. Both left and right side are anti-symmetric.