Despite the progress on 3D point cloud deep learning, most prior works focus on learning features that are invariant to translation and point permutation, and very limited efforts have been devoted for rotation invariant property. Several recent studies achieve rotation invariance at the cost of lower accuracies. In this work, we close this gap by proposing a novel yet effective rotation invariant architecture for 3D point cloud classification and segmentation. Instead of traditional pointwise operations, we construct local triangle surfaces to capture more detailed surface structure, based on which we can extract highly expressive rotation invariant surface properties which are then integrated into an attention-augmented convolution operator named RISurAAConv to generate refined attention features via self-attention layers. Based on RISurAAConv we build an effective neural network for 3D point cloud analysis that is invariant to arbitrary rotations while maintaining high accuracy. We verify the performance on various benchmarks with supreme results obtained surpassing the previous state-of-the-art by a large margin. We achieve 95.3% (+4.3%) on ModelNet40, 92.6% (+12.3%) on ScanObjectNN, and 96.4% (+7.0%), 87.6% (+13.0%), 88.7%} (+7.7%) respectively on the three categories of FG3D dataset for fine-grained classification task and achieve 81.5% (+1.0%) mIoU on ShapeNet for segmentation task, respectively. The code and models will be released upon publication.