Progress in anatomical 3D shape classification is limited by the complexity of mesh data and the lack of standardized benchmarks, highlighting the need for robust learning methods and reproducible evaluation. We introduce two key steps toward clinically and benchmark-ready anatomical shape classification via self-supervised graph autoencoding. We propose Precomputed Structural Pooling (PSPooling), a non-learnable mesh pooling operator designed for efficient and structure-preserving graph coarsening in 3D anatomical shape analysis. PSPooling precomputes node correspondence sets based on geometric proximity, enabling parallelizable and reversible pooling and unpooling operations with guaranteed support structure. This design avoids the sparsity and reconstruction issues of selection-based methods and the sequential overhead of edge contraction approaches, making it particularly suitable for high-resolution medical meshes. To demonstrate its effectiveness, we integrate PSPooling into a self-supervised graph autoencoder that learns anatomy-aware representations from unlabeled surface meshes. We evaluate the downstream benefits on MedShapeNet19, a new curated benchmark dataset we derive from MedShapeNet, consisting of 19 anatomical classes with standardized training, validation, and test splits. Experiments show that PSPooling significantly improves reconstruction fidelity and classification accuracy in low-label regimes, establishing a strong baseline for medical 3D shape learning. We hope that MedShapeNet19 will serve as a widely adopted benchmark for anatomical shape classification and further research in medical 3D shape analysis. Access the complete codebase, model weights, and dataset information here: https://github.com/TomasKrsicka/MedShapeNet19-PSPooling.