A system of two coupled ensembles of phase oscillators can follow different routes to interensemble synchronization. Following a short report of our preliminary results [Phys. Rev. E 78, 025201(R) (2008)], we present a more detailed study of the effects of coupling, noise, and phase asymmetries in coupled phase-oscillator ensembles. We identify five distinct synchronization regions and routes to synchronization that are characteristic of the coupling asymmetry. We show that noise asymmetry induces effects similar to that of coupling asymmetry when the latter is absent. We also find that phase asymmetry controls the probability of occurrence of particular routes to synchronization. Our results suggest that asymmetry plays a crucial role in controlling synchronization within and between oscillator ensembles, and hence that its consideration is vital for modeling real life problems
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