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Recently, many deep neural networks were designed to process 3D point clouds, but a common drawback is that rotation invariance is not ensured, leading to poor generalization to arbitrary orientations. In this paper, we introduce a new low-level purely rotation-invariant representation to replace common 3D Cartesian coordinates as the network inputs. Also, we present a network architecture to embed these representations into features, encoding local relations between points and their neighbors, and the global shape structure. To alleviate inevitable global information loss caused by the rotation-invariant representations, we further introduce a region relation convolution to encode local and non-local information. We evaluate our method on multiple point cloud analysis tasks, including shape classification, part segmentation, and shape retrieval. Experimental results show that our method achieves consistent, and also the best performance, on inputs at arbitrary orientations, compared with the state-of-the-arts.