New S-dualities in N=2 supersymmetric SU(2) x SU(2) gauge theory

New S-dualities in a scale invariant N = 2 gauge theory with SU(2) x SU(2) gauge group are derived from embeddings of the theory in two different larg er asymptotically free theories. The true coupling space of the scale invar iant theory is a 20-fold identification of the coupling space found in the M- and string-theory derivations of the low energy effective action, implyi ng a larger S-duality group. Also, this coupling space is different from th e naively expected direct product of two SL(2; Z) fundamental domains, as i t contains a different topology of fixed points.