  Skew-gentle algebras are skew-group algebras of gentle algebras equipped with
a certain $\Z_2$-action. Building on the bijective correspondence between
gentle algebras and dissected surfaces, we obtain in this paper a bijection
between skew-gentle algebras and certain dissected orbifolds that admit a
double cover.
  We prove the compatibility of the $\Z_2$-action on the double cover with the
skew-group algebra construction. This allows us to investigate the derived
equivalence relation between skew-gentle algebras in geometric terms: We
associate to each skew-gentle algebra a line field on the orbifold, and on its
double cover, and interpret different kinds of derived equivalences of
skew-gentle algebras in terms of diffeomorphisms respecting the homotopy class
of the line fields associated to the algebras.
