High-order-dispersion-induced dispersive waves emitted by dissipative Kerr solitons are frequently observed in microresonator frequency comb generation. Also known as soliton Cherenkov radiation, this type of dispersive wave plays a critical role in comb spectrum broadening as well as the formation of soliton bound states. Here, we report the experimental observation of symmetry breaking in the group velocity of counterpropagating solitons in a crystalline microresonator. Induced by the polychromatic Cherenkov radiation, soliton bound states are formed, showing different group velocities with different spatiotemporal separations between constituent solitons. By bidirectionally pumping the microresonator with laser fields of equal power and frequency, we demonstrate the degeneracy lifting of repetition rates of the counterpropagating solitons. Our work not only shines new light on the impact of dispersive waves in nonlinear cavities, but also introduces a novel approach to develop compact dual-comb spectrometers.