Abstract. Let D be a digraph. Its inverse digraph D-1 is obtained by reversing the orientation of all arcs of D. We show that the domination numbers of D and D-1 may be different if D is a Cayley digraph. The smallest groups admitting such digraphs are the alternating and dihedral groups on 12 elements. We generalize the dihedral group example to all dihedral groups on at least 12 elements. Analogous results are obtained for the total domination number.