Recent works on Kerr frequency combs demonstrated the possibility to simultaneously generate multiple soliton states with different group velocities in a single microresonator, which could be beneficial for many applications utilizing dual-comb systems. In this work we demonstrate a dual-comb configuration in a single crystalline microresonator by monochromatically pumping counter-propagating solitons in the same spatial mode with equal powers. Moreover, we demonstrate experimentally and through simulations the key role of Cherenkov radiation interferences on the repetition rate of a multi-soliton state. This result not only shines new light on the impact of dispersive waves on dissipative Kerr solitons but also introduces a novel approach to develop coherent dual-comb spectrometer based on microcombs.