Kac's `hear the shape of a drum" problem is the extent to which a plane domain is determined by its Dirichlet eigenvalues. I.e. is the map from domains to their spectra 1-1. We show that if the spectrum map is restricted to analytic plane domains with one up down symmetry (and an axis length fixed), then it is one-one. I.e. you can determine such a domain from its eigenvalues among other such domains. In joint work with Hamid Hezari, we also give a generalization to higher dimensions.