The flip map on the full 2 shift is the map that exchanges symbols. Flip commuting maps have been thought to be rare. We will discuss the set of automorphisms of the full 2-shift that commute with the flip map, including giving the first set of explicit examples and a characterization in terms of the quotient shift. We will also discuss the quotient shifts of strictly order \(n\) automorphisms on 1-sided shifts of finite type.