We show, using semiclassical methods, that the spectral correlations of quantum maps undergo a rapid transition from one universality class to another when the symmetry of the system is gradually broken. This transition turns out to be governed by the same parametrization as in the theory of random matrices. The theory is quantitatively verified for the kicked rotor system. We also provide an explicit substantiation of the random matrix hypothesis, namely, that in the symmetry adapted basis the symmetry violating part is random.