Point cloud registration is one of the basic research hotspots in the field of 3D reconstruction. Although many previous studies have made great progress, the registration of rock point clouds remains an ongoing challenge, due to the complex surface, arbitrary shape, and high resolution of rock masses. To overcome these challenges, a novel registration method for rock point clouds, based on local invariants, is proposed in this paper. First, to handle the massive point clouds, a point of interest filtering method based on a sum vector is adopted to reduce the number of points. Second, the remaining points of interest are divided into several cluster point sets and the centroid of each cluster is calculated. Then, we determine the correspondence between the original point cloud and the target point cloud by proving the inherent similarity (using the trace of the covariance matrix) of the remaining point sets. Finally, the rotation matrix and translation vector are calculated, according to the corresponding centroids, and a correction method is used to adjust the positions of the centroids. To illustrate the superiority of our method, in terms of accuracy and efficiency, we conducted experiments on multiple datasets. The experimental results show that the method has higher accuracy (about ten times) and efficiency than similar existing methods.