 Isospectral hyperbolic surfaces have matching geodesics by Peter G. Doyle and Juan Pablo Rossetti

Journal full text: nyjm.albany.edu/j/2008/14-7.pdf

New York Journal of Mathematics 14 (2008), pages 193–204. Theorem 1 distinguishes orientation-preserving and orientation-reversing closed geodesics for nonorientable surfaces. Sections 2–3 explain their different trace weights.