  Rauzy-type dynamics are group actions on a collection of combinatorial
objects. The first and best known example (the Rauzy dynamics) concerns an
action on permutations, associated to interval exchange transformations (IET)
for the Poincar\'e map on compact orientable translation surfaces. The
equivalence classes on the objects induced by the group action have been
classified by Kontsevich and Zorich, and by Boissy through methods involving
both combinatorics algebraic geometry, topology and dynamical systems. Our
first paper proposed an ad hoc combinatorial proof of this classification. In
this paper we define a general method, called the labelling method, which
allows one to classify Rauzy-type dynamics in a much more systematic way. We
apply the method to the Rauzy dynamics and obtain a second combinatorial proof
of the classification.
