  In this paper we present a topological framework for studying signed
permutations and their reversal distance. As a result we can give an
alternative approach and interpretation of the Hannenhalli-Pevzner formula for
the reversal distance of signed permutations. Our approach utlizes the
Poincar\'e dual, upon which reversals act in a particular way and obsoletes the
notion of "padding" of the signed permutations. To this end we construct a
bijection between signed permutations and an equivalence class of particular
fatgraphs, called $\pi$-maps, and analyze the action of reversals on the
latter. We show that reversals act via either slicing, gluing or half-flipping
of external vertices, which implies that any reversal changes the topological
genus by at most one. Finally we revisit the Hannenhalli-Pevzner formula
employing orientable and non-orientable, irreducible, $\pi$-maps.
