An orientation reversing involution s of a topological compact
genus g, g>2, surface S induces an antiholomorphic involution
of the Teichmüller space of of genus g Riemann surfaces. Two such
involutions s* and t* are conjugate in the
mapping class group if and only if the corresponding
orientation reversing involutions s and t
of S are conjugate in the automorphism group of S. This
is equivalent to saying that the quotient surfaces
This result is a simple fact that follows from Royden's theorem stating that the the mapping class group is the full group of holomorphic automorphisms of the Teichmüller space (g>2).
be two real structures that are not conjugate in the mapping class
group. In this paper we construct a real analytic
diffeomorphism d: Tg -> Tg such that
This mapping d is a product of full and half Dehn-twists around certain simple closed curves on the surface S.
This result has applications to the moduli spaces of real algebraic curves.