  We construct an explicit bijection between bipartite pointed maps of an
arbitrary surface $\mathbb{S}$, and specific unicellular blossoming maps of the
same surface. Our bijection gives access to the degrees of all the faces, and
distances from the pointed vertex in the initial map. The main construction
generalizes recent work of the second author which covered the case of an
orientable surface.
  Our bijection gives rise to a first combinatorial proof of a parametric
rationality result concerning the bivariate generating series of maps of a
given surface with respect to their numbers of faces and vertices. In
particular, it provides a combinatorial explanation of the structural
difference between the aforementioned bivariate parametric generating series in
the case of orientable and non-orientable maps.
