We analyze the synchronization dynamics of the thermodynamically large systems of globally coupled phase oscillators with bimodal distribution of frequencies with asymmetry between two distribution components. The dynamics and the transitions between various synchronous and asynchronous regimes are shown to be very sensitive to the degree of the asymmetry whereas the scenario of the symmetry breaking is universal and does not depend on the particular way to introduce asymmetry, be it the non-equal populations of modes in bimodal distribution, the delay of the Kuramoto-Sakaguchi model, different values of the coupling constants, or the different noise levels for oscillators in different distribution modes. In particular, we found that even small asymmetry may stabilize the stationary partially synchronized state, and this may happen even for arbitrarily large frequency difference between two distribution modes (oscillator subgroups). This effect also results in the new type of bistability in the system when two stationary partially synchronized states coexist: one with large level of global synchronization and synchronization parity between two subgroups and another with lower synchronization where the one subgroup is dominant, having higher internal (subgroup) synchronization level and enforcing its oscillation frequency on the second subgroup.