Slit-slide-sew bijections for constellations and quasiconstellations

We extend so-called slit-slide-sew bijections to constellations and quasiconstellations. We present an involution on the set of hypermaps given with an orientation, one distinguished corner, and one distinguished edge leading away from the corner while oriented in the given orientation. This involution reverts the orientation, exchanges the distinguished corner with the distinguished edge in some sense, slightly modifying the degrees of the incident faces in passing, while keeping all the other faces intact. The involution specializes into a bijection interpreting combinatorial identities and allows to recover the counting formula for constellations or quasiconstellations with a given face degree distribution.