Autonomous vehicles and robots rely on accurate odometry estimation in GPS-denied environments. While LiDARs and cameras struggle under extreme weather, 4D mmWave radar emerges as a robust alternative with all-weather operability and velocity measurement. In this paper, we introduce Equi-RO, an equivariant network-based framework for 4D radar odometry. Our algorithm pre-processes Doppler velocity into invariant node and edge features in the graph, and employs separate networks for equivariant and invariant feature processing. A graph-based architecture enhances feature aggregation in sparse radar data, improving inter-frame correspondence. Experiments on an open-source dataset and a self-collected dataset show Equi-RO outperforms state-of-the-art algorithms in accuracy and robustness. Overall, our method achieves 10.7% and 13.4% relative improvements in translation and rotation accuracy, respectively, compared to the best baseline on the open-source dataset.