  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. We will also show that
self-inverse permutations are Lagrangian, meaning any suspension has its
vertical cycles span a Lagrangian subspace in homology. This will simplify the
proof of a lemma in a work by G. Forni. W. A. Veech proved a bound on the
number of distinct ergodic probability measures for a given minimal interval
exchange transformation. We verify that this bound is sharp by construcing
examples in each Rauzy Class.
