Domain adaptive LiDAR point cloud segmentation aims to learn an effective target segmentation model from labelled source data and unlabelled target data, which has attracted increasing attention in recent years due to the difficulty in point-cloud annotation. It remains a very open research challenge as point clouds of different domains often have clear distribution discrepancies with variations in LiDAR sensor configurations, environmental conditions, occlusions, etc. We design a simple yet effective spatial consistency training framework that can learn superior domain-invariant feature representations from unlabelled target point clouds. The framework exploits three types of spatial consistency, namely, geometric-transform consistency, sparsity consistency, and mixing consistency which capture the semantic invariance of point clouds with respect to viewpoint changes, sparsity changes, and local context changes, respectively. With a concise mean teacher learning strategy, our experiments show that the proposed spatial consistency training outperforms the state-of-the-art significantly and consistently across multiple public benchmarks. Code is available at https://github.com/xiaoaoran/sct-uda.