In this paper we exhibit two interlocking sequences in order to study the group ε(X×Y) of based homotopy classes of based self homotopy equivalences of a productX×Yof topological spacesXandY.These sequences have complementary features, and the interconnectedness facilitates computation. The sequences give new results about ε(X×Y) and unify, generalize, and in some cases correct, existing results in the literature about this group. New results include calculations on the group of self equivalences of a product of suspensions (with applications to a product of Moore spaces, includingplocalized spheres); some progress in the computation of non-simply-connected rank 2H-spaces (posed as an open problem in [Ka;problem 5]); and a universal description of ε(Sm×Sn) forn>m≥ 1 as a semidirect product. This last situation includes an example that is not a semidirect product when decomposed in the only other way known (see section 4).