Three-dimensional point cloud registration is an important field in computer vision. Recently, due to the increasingly complex scenes and incomplete observations, many partial-overlap registration methods based on overlap estimation have been proposed. These methods heavily rely on the extracted overlapping regions with their performances greatly degraded when the overlapping region extraction underperforms. To solve this problem, we propose a partial-to-partial registration network (RORNet) to find reliable overlapping representations from the partially overlapping point clouds and use these representations for registration. The idea is to select a small number of key points called reliable overlapping representations from the estimated overlapping points, reducing the side effect of overlap estimation errors on registration. Although it may filter out some inliers, the inclusion of outliers has a much bigger influence than the omission of inliers on the registration task. The RORNet is composed of overlapping points' estimation module and representations' generation module. Different from the previous methods of direct registration after extraction of overlapping areas, RORNet adds the step of extracting reliable representations before registration, where the proposed similarity matrix downsampling method is used to filter out the points with low similarity and retain reliable representations, and thus reduce the side effects of overlap estimation errors on the registration. Besides, compared with previous similarity-based and score-based overlap estimation methods, we use the dual-branch structure to combine the benefits of both, which is less sensitive to noise. We perform overlap estimation experiments and registration experiments on the ModelNet40 dataset, outdoor large scene dataset KITTI, and natural data Stanford Bunny dataset. The experimental results demonstrate that our method is superior to other partial registration methods. Our code is available at https://github.com/superYuezhang/RORNet.