Optical microresonators offer a highly-attractive new platform for the generation of optical frequency combs. Recently, several groups have been able to demonstrate the generation of dual-frequency combs in a single microresonator driven by two optical pumps. This opens the possibility for microresonator-based dual-comb systems suitable for measurement applications such as spectroscopy, ranging and imaging. Key to the performance of these systems are the parameters of the radio-frequency comb spectrum that arises from the interference of the two optical combs. In this work, we present a simple mechanism to enable the discrete fine-tuning of these parameters by driving the two optical combs with optical pumps with different azimuthal mode numbers. The mechanism consists of tuning the difference in azimuthal mode number between the two pumps by selection of the pumps' frequencies. We are able to implement this technique when the two counter-propagating pumps are set to drive resonances of the same spatial mode family, as well as different mode families. In each case, we experimentally observe ∼1 MHz of discrete tunability in the line-spacing of the radio-frequency comb as the frequency offset between the two pumps is scanned between 0 to 80 free-spectral-ranges.