Our eventual goal is to see that if X is any linear order that isomorphic to its cube, then X^{\omega} has a parity-reversing automorphism. Then by the results of last week, X will also be isomorphic to its square. This week, I will describe a method for building partial parity-reversing automorphisms on any A^{\omega}, and give structural conditions on A under which these partial automorphisms can be stitched together to get a full p.r.a. We will see in particular that if A is countable, then A^{\omega} has a p.r.a.