Spinning, frequency-modulated continuous-wave (FMCW) radar has been gaining popularity for autonomous vehicle navigation. The spinning radar is chosen over the more classic automotive `fixed' radar as it is able to capture the full 360 degree field of view without requiring multiple sensors and extensive calibration. However, commercially available spinning radar systems have not previously had the ability to extract radial velocities due to the lack of repeated measurements in the same direction and fundamental hardware setup. A new firmware upgrade now makes it possible to alternate the modulation of the radar signal between azimuths. In this paper, we first present a way to use this alternating modulation to extract radial Doppler velocity measurements from single raw radar intensity scans. We then incorporate these measurements in two different modern odometry pipelines and evaluate them in progressively challenging autonomous driving environments. We show that using Doppler velocity measurements enables our odometry to continue functioning at state-of-the-art even in severely geometrically degenerate environments.