Consider a finite group $G$ acting on a Riemann surface $S$, and
the associated branched Galois cover $\pi_G:S \to Y=S/G$. We
introduce the concept of \emph{geometric signature} for the action
of $G$, and we show that it captures much information: the
geometric structure of the lattice of intermediate covers, the
isotypical decomposition of the rational representation of the
group $G$ acting on the Jacobian variety $JS$ of $S$, and the
dimension of the subvarieties of the isogeny decomposition of
$JS$. We also give a version of Riemann's existence theorem,
adjusted to the present setting.