Abstract. Let D be a digraph. By γ(D) we denote the domination number of D and by D^- we denote a digraph obtained by reversing all the arcs of D. We prove that for every d>2 and k>1 there exists a simple strongly connected d-regular digraph D_d,k such that γ(D^-_d,k)-γ(D_d,k)=k. Analogous theorem is obtained for total domination number provided that d>3. To obtain the digraph D_d,k we lift a small digraph (with loops) first in Z₃ to obtain a simple strongly connected and d-regular digraph, and then we lift it in Z_2k.