Extracting distinctive, robust, and general 3D local features is essential to downstream tasks such as point cloud registration. However, existing methods either rely on noise-sensitive handcrafted features, or depend on rotation-variant neural architectures. It remains challenging to learn robust and general local feature descriptors for surface matching. In this paper, we propose a new, simple yet effective neural network, termed SpinNet, to extract local surface descriptors which are rotation-invariant whilst sufficiently distinctive and general. A Spatial Point Transformer is first introduced to embed the input local surface into an elaborate cylindrical representation (SO(2) rotation-equivariant), further enabling end-to-end optimization of the entire framework. A Neural Feature Extractor, composed of point-based and 3D cylindrical convolutional layers, is then presented to learn representative and general geometric patterns. An invariant layer is finally used to generate rotation-invariant feature descriptors. Extensive experiments on both indoor and outdoor datasets demonstrate that SpinNet outperforms existing state-of-the-art techniques by a large margin. More critically, it has the best generalization ability across unseen scenarios with different sensor modalities.