Classification of the orientation reversing homeomorphisms of finite order of surfaces

Let S be a topological, orientable, closed surface. Two (self)-homeomorphisms Φ1 and Φ2 of S are conjugate if there is an orientation preserving homeomorphism h such that Φ1 = h−1Φ2h. In this paper we classify up to conjugation the orientation reversing homeomorphisms with finite order greater than two. The action of a homeomorphism Φ on a surface S produces an orbifold structure on the quotient S/Φ. Our classification is obtained using such orbifold structures.