Rotation-invariant recognition of shapes is a common challenge in computer vision. Recent approaches have significantly improved the accuracy of rotation-invariant recognition by encoding the rotational invariance of shapes as hand-crafted image features and introducing deep neural networks. However, the methods based on pixels have too much redundant information, and the critical geometric information is prone to early leakage, resulting in weak rotation-invariant recognition of fine-grained shapes. In this paper, we reconsider the shape recognition problem from the perspective of contour points rather than pixels. We propose an anti-noise rotation-invariant convolution module based on contour geometric aware for fine-grained shape recognition. The module divides the shape contour into multiple local geometric regions (LGA), where we implement finer-grained rotation-invariant coding in terms of point topological relations. We provide a deep network composed of five such cascaded modules for classification and retrieval experiments. The results show that our method exhibits excellent performance in rotation-invariant recognition of fine-grained shapes. In addition, we demonstrate that our method is robust to contour noise and the rotation centers.