We study the (delta;D) and (delta;N) problems for double-step digraphs considering the unilateral distance, which is the minimum between the distance in the digraph and the distance in its converse digraph, the latter obtained by changing the directions of all the arcs. The first problem consists of maximizing the number of vertices N of a digraph, given the maximum degree and the unilateral diameter D , whereas the second one (somehow dual of the first) consists of minimizing the unilateral diameter given the maximum degree and the number of vertices. We solve the first problem for every value of the unilateral diameter and the second one for infinitely many values of the number of vertices. Moreover, we compute the mean unilateral distance of the digraphs in the families considered.