This paper introduces a robust unsupervised SE(3) point cloud registration method that operates without requiring point correspondences. The method frames point clouds as functions in a reproducing kernel Hilbert space (RKHS), leveraging SE(3) equivariant features for direct feature space registration. A novel RKHS distance metric is proposed, offering reliable performance amidst noise, outliers, and asymmetrical data. An unsupervised training approach is introduced to effectively handles limited ground truth data, facilitating adaptation to real datasets. The proposed method outperforms traditional supervised methods in terms of registration accuracy on both synthetic (ModelNet) and real-world (ETH-3D) noisy, outlier-rich datasets, marking the first instance of successful real RGB-D odometry data registration using an equivariant method. The code will be made available upon publication.