We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by an element of finite order of the normalizer of the associated complex reflection group. We give a parametrization à la Harish-Chandra of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which is the theme of another paper.