  A unicellular map is the embedding of a connected graph in a surface in such
a way that the complement of the graph is a topological disk. In this paper we
present a bijective link between unicellular maps on a non-orientable surface
and unicellular maps of a lower topological type, with distinguished vertices.
From that we obtain a recurrence equation that leads to (new) explicit counting
formulas for non-orientable unicellular maps of fixed topology. In particular,
we give exact formulas for the precubic case (all vertices of degree 1 or 3),
and asymptotic formulas for the general case, when the number of edges goes to
infinity. Our strategy is inspired by recent results obtained by the second
author for the orientable case, but significant novelties are introduced: in
particular we construct an involution which, in some sense, "averages" the
effects of non-orientability.
