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Rauzy-type dynamics are group actions on a collection of combinatorialobjects. The first and best known example concerns an action on permutations,associated to interval exchange transformations (IET) for the Poincar\'e map oncompact orientable translation surfaces. The equivalence classes on the objectsinduced by the group action are related to components of the moduli spaces ofAbelian differentials with prescribed singularities, and, in two variants ofthe problem, have been classified by Kontsevich and Zorich, and by Boissy,through methods involving both combinatorics and algebraic geometry.  We provide here a purely combinatorial proof of both classification theorems,and in passing establish a few previously unnoticed features. As will be shownelsewhere, our methods extend also to other Rauzy-type dynamics, both onlabeled and unlabeled structures. Some of these dynamics have a geometricalinterpretation (e.g., matchings, related to IET on non-orientable surfaces),while some others do not have one so far.

A combinatorial approach to Rauzy-type dynamics I: permutations and the Kontsevich--Zorich--Boissy classification theorem