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Rauzy-type dynamics are group actions on a collection of combinatorialobjects. The first and best known example (the Rauzy dynamics) concerns anaction on permutations, associated to interval exchange transformations (IET)for the Poincar\'e map on compact orientable translation surfaces. Theequivalence classes on the objects induced by the group action have beenclassified by Kontsevich and Zorich, and by Boissy through methods involvingboth combinatorics algebraic geometry, topology and dynamical systems. Ourfirst paper proposed an ad hoc combinatorial proof of this classification. Inthis paper we define a general method, called the labelling method, whichallows one to classify Rauzy-type dynamics in a much more systematic way. Weapply the method to the Rauzy dynamics and obtain a second combinatorial proofof the classification.

A combinatorial approach to Rauzy-type dynamics II: the labelling method and a second proof of the KZB classification theorem