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We construct an explicit bijection between bipartite pointed maps of anarbitrary surface $\mathbb{S}$, and specific unicellular blossoming maps of thesame surface. Our bijection gives access to the degrees of all the faces, anddistances from the pointed vertex in the initial map. The main constructiongeneralizes recent work of the second author which covered the case of anorientable surface.  Our bijection gives rise to a first combinatorial proof of a parametricrationality result concerning the bivariate generating series of maps of agiven surface with respect to their numbers of faces and vertices. Inparticular, it provides a combinatorial explanation of the structuraldifference between the aforementioned bivariate parametric generating series inthe case of orientable and non-orientable maps.

Blossoming bijection for bipartite pointed maps and parametric rationality of general maps of any surface