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Thanks to works by M. Kontsevich and A. Zorich followed by C. Boissy, we havea classification of all Rauzy Classes of any given genus. It follows from theseworks that Rauzy Classes are closed under the operation of inverting thepermutation. In this paper, we shall prove the existence of self-inversepermutations in every Rauzy Class by giving an explicit construction of such anelement satisfying the sufficient conditions. We will also show thatself-inverse permutations are Lagrangian, meaning any suspension has itsvertical cycles span a Lagrangian subspace in homology. This will simplify theproof of a lemma in a work by G. Forni. W. A. Veech proved a bound on thenumber of distinct ergodic probability measures for a given minimal intervalexchange transformation. We verify that this bound is sharp by construcingexamples in each Rauzy Class.

Self-Inverses, Lagrangian Permutations and Minimal Interval Exchange Transformations with Many Ergodic Measures