  In this paper we introduce a new invariant for the action of a finite group
$G$ on a compact complex curve of genus $g$. With the aid of this invariant we
achieve the classification of the components of the moduli space of curves with
an effective action by the dihedral group $D_n$. This invariant has been used
in the meanwhile by the authors in order to extend the genus stabilization
result of Livingston and Dunfield and Thurston to the ramified case. This new
version contains an appendix clarifying the correspondence between the above
components and the image loci in the moduli space M_g (classifying when two
such components have the same image).
