Point cloud registration is a foundational task crucial for 3D alignment and reconstruction applications. While both traditional and learning-based registration approaches have achieved significant success, the leverage of intrinsic symmetry within input point cloud data often receives insufficient attention, which prohibits the model from sample leveraging and learning efficiency, leading to an increase in data size and model complexity. To address these challenges, we propose a dedicated graph neural network model embedded with a built-in Spherical Euclidean 3D equivariance property achieved through SE(3) node features and message passing equivariance. Pairwise input feature embeddings are derived from sparsely downsampled input point clouds, each with several orders of magnitude less point than two raw input point. These embeddings form a rigidity graph capturing spatial relationships, which is subsequently pooled into global features followed by cross-attention mechanisms, and finally decoded into the regression pose between point clouds. Experiments conducted on the 3DMatch and KITTI datasets exhibits the compelling and distinctive performance of our model compared to state-of-the-art approaches. By harnessing the equivariance properties inherent in the data, our model exhibits a notable reduction in the required input points during training when compared to existing approaches relying on dense input points. Moreover, our model eliminates the necessity for the removal or modeling of outliers present in dense input points, which simplifies the pre-processing pipeline of point cloud.