  We present an extensive study of the Eulerian distribution on the set of self
evacuated involutions, namely, involutions corresponding to standard Young
tableaux that are fixed under the Sch$\ddot{\textrm{u}}$tzenberger map. We find
some combinatorial properties for the generating polynomial of such
distribution, together with an explicit formula for its coefficients.
Afterwards, we carry out an analogous study for the subset of self evacuated
involutions without fixed points.
