  The main result of this paper is that one cannot hear orientability of a
surface with boundary. More precisely, we construct two isospectral flat
surfaces with boundary with the same Neumann spectrum, one orientable, the
other non-orientable. For this purpose, we apply Sunada's and Buser's methods
in the framework of orbifolds. Choosing a symmetric tile in our construction,
and adapting a folklore argument of Fefferman, we also show that the surfaces
have different Dirichlet spectra. These results were announced in the {\it C.
R. Acad. Sci. Paris S\'er. I Math.}, volume 320 in 1995, but the full proofs so
far have only circulated in preprint form.
