One can't hear orientability of surfaces.

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.