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We prove that, under some generic non-degeneracy assumptions, real analytic,centrally symmetric plane domains are determined by their Dirichlet (resp.Neumann) spectra. We prove that the conditions are open-dense for real analyticconvex domains. The proof is parallel to the proof that up/down symmetricdomains are spectrally determined. One step is to use a Maslov indexcalculation to show that the second derivative of the defining function of acentrally symmetric domain at the endpoints of a bouncing ball orbit is aspectral invariant. This is also true for up/down symmetric domains, removingan assumption from the proof in that case.

Centrally symmetric analytic plane domains are spectrally determined in this class