Abstract Current neural networks for 3D object recognition are vulnerable to 3D rotation. Existing works mostly rely on massive amounts of rotation-augmented data to alleviate the problem, which lacks solid guarantee of the 3D rotation invariance. In this paper, we address the issue by introducing a novel point cloud representation that can be mathematically proved rigorously rotation-invariant, i.e., identical point clouds in different orientations are unified as a unique and consistent representation. Moreover, the proposed representation is conditional information-lossless, because it retains all necessary information of point cloud except for orientation information. In addition, the proposed representation is complementary with existing network architectures for point cloud and fundamentally improves their robustness against rotation transformation. Finally, we propose a deep hierarchical cluster network called ClusterNet to better adapt to the proposed representation. We employ hierarchical clustering to explore and exploit the geometric structure of point cloud, which is embedded in a hierarchical structure tree. Extensive experimental results have shown that our proposed method greatly outperforms the state-of-the-arts in rotation robustness on rotation-augmented 3D object classification benchmarks.