3D anomaly detection (AD) is a crucial task in computer vision, aiming to identify anomalous points or regions from point cloud data. However, existing methods may encounter challenges when handling point clouds with changes in orientation and position because the resulting features may vary significantly. To address this problem, we propose a novel Rotationally Invariant Features (RIF) framework for 3D AD. Firstly, to remove the adverse effect of variations on point cloud data, we develop a Point Coordinate Mapping (PCM) technique, which maps each point into a rotationally invariant space to maintain consistency of representation. Then, to learn robust and discriminative features, we design a lightweight Convolutional Transform Feature Network (CTF-Net) to extract rotationally invariant features for the memory bank. To improve the ability of the feature extractor, we introduce the idea of transfer learning to pre-train the feature extractor with 3D data augmentation. Experimental results show that the proposed method achieves the advanced performance on the Anomaly-ShapeNet dataset, with an average P-AUROC improvement of 17.7\%, and also gains the best performance on the Real3D-AD dataset, with an average P-AUROC improvement of 1.6\%. The strong generalization ability of RIF has been verified by combining it with traditional feature extraction methods on anomaly detection tasks, demonstrating great potential for industrial applications.