  Rauzy Classes and Extended Rauzy Classes are equivalence classes of
permutations that arise when studying Interval Exchange Transformations. In
2003, Kontsevich and Zorich classified Extended Rauzy Classes by using data
from Translation Surfaces, which are associated to IET's thanks to the Zippered
Rectangle Construction of Veech from 1982. In 2009, Boissy finalized the
classification of Rauzy Classes also using information from Translation
Surfaces. We present in this paper specialized moves in (Extended) Rauzy
Classes that allows us to prove the sufficiency and necessity in the previous
classification theorems. These results provide a complete, and purely
combinatorial, proof of these known results. We end with some general
statements about our constructed move.
