Point cloud classification is critical for three-dimensional scene understanding. However, in real-world scenarios, depth cameras often capture partial, single-view point clouds of objects with different poses, making their accurate classification a challenge. In this paper, we propose a novel point cloud classification network that captures the detailed spatial structure of objects by constructing tetrahedra, which is different from point-wise operations. Specifically, we propose a RISpaNet block to extract rotation-invariant features. A rotation-invariant property generation module is designed in RISpaNet for constructing rotation-invariant tetrahedron properties (RITPs). Meanwhile, a multi-scale pooling module and a hybrid encoder are used to process RITPs to generate integrated rotation-invariant features. Further, for single-view point clouds, a complete point cloud auxiliary branch and a part-whole correlation module are jointly employed to obtain complete point cloud features from partial point clouds. Experimental results show that this network performs better than other state-of-the-art methods, evaluated on four public datasets. We achieved an overall accuracy of 94.7% (+2.0%) on ModelNet40, 93.4% (+5.9%) on MVP, 94.7% (+6.3%) on PCN and 94.8% (+1.7%) on ScanObjectNN. Our project website is https://luxurylf.github.io/RISpaNet_project/.