A renaissance in radar-based sensing for mobile robotic applications is underway. Compared to cameras or lidars, millimetre-wave radars have the ability to `see' through thin walls, vegetation, and adversarial weather conditions such as heavy rain, fog, snow, and dust. In this paper, we propose a novel SE(2) odometry approach for spinning frequency-modulated continuous-wave radars. Our method performs scan-to-local-map registration of the incoming radar data in a direct manner using all the radar intensity information without the need for feature or point cloud extraction. The method performs locally continuous trajectory estimation and accounts for both motion and Doppler distortion of the radar scans. If the radar possesses a specific frequency modulation pattern that makes radial Doppler velocities observable, an additional Doppler-based constraint is formulated to improve the velocity estimate and enable odometry in geometrically feature-deprived scenarios (e.g., featureless tunnels). Our method has been validated on over 250km of on-road data sourced from public datasets (Boreas and MulRan) and collected using our automotive platform. With the aid of a gyroscope, it outperforms state-of-the-art methods and achieves an average relative translation error of 0.26% on the Boreas leaderboard. When using data with the appropriate Doppler-enabling frequency modulation pattern, the translation error is reduced to 0.18% in similar environments. We also benchmarked our algorithm using 1.5 hours of data collected with a mobile robot in off-road environments with various levels of structure to demonstrate its versatility. Our real-time implementation is publicly available: https://github.com/utiasASRL/dro.