  Thanks to works by M. Kontsevich and A. Zorich followed by C. Boissy, we have
a classification of all Rauzy Classes of any given genus. It follows from these
works that Rauzy Classes are closed under the operation of inverting the
permutation. In this paper, we shall prove the existence of self-inverse
permutations in every Rauzy Class by giving an explicit construction of such an
element satisfying the sufficient conditions. As a corollary, we will give
another proof that every Rauzy Class is closed under taking inverses. In the
case of generalized permutations, generalized Rauzy Classes have been
classified by works of M. Kontsevich, H. Masur and J. Smillie, E. Lanneau, and
again C. Boissy. We state the definition of self-inverse for generalized
permutations and prove a necessary and sufficient condition for a generalized
Rauzy Class to contain self-inverse elements.

Dynamical Systems