Dense 3D shape correspondence remains a central challenge in computer vision and graphics as many deep learning approaches still rely on intermediate geometric features or handcrafted descriptors, limiting their effectiveness under non-isometric deformations, partial data, and non-manifold inputs. To overcome these issues, we introduce RINO, an unsupervised, rotation-invariant dense correspondence framework that effectively unifies rigid and non-rigid shape matching. The core of our method is the novel RINONet, a feature extractor that integrates vector-based SO(3)-invariant learning with orientation-aware complex functional maps to extract robust features directly from raw geometry. This allows for a fully end-to-end, data-driven approach that bypasses the need for shape pre-alignment or handcrafted features. Extensive experiments show unprecedented performance of RINO across challenging non-rigid matching tasks, including arbitrary poses, non-isometry, partiality, non-manifoldness, and noise.