Building on our characterization from last week of the orders X that are isomorphic to AX, we characterize those X that are isomorphic to AAX. We then write down a condition -- namely, the existence of a parity-reversing automorphism (p.r.a.) for the countable power of A -- under which the implication ``AAX = X implies AX = X" holds. In future talks, we will show that if X is isomorphic to its cube then the countable power of X has a p.r.a., and hence X is isomorphic to its square.