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Unsupervised point cloud completion aims at estimating the corresponding complete point cloud of a partial point cloud in an unpaired manner. It is a crucial but challenging problem since there is no paired partial-complete supervision that can be exploited directly. In this work, we propose a novel framework, which learns a unified and structured latent space that encoding both partial and complete point clouds. Specifically, we map a series of related partial point clouds into multiple complete shape and occlusion code pairs and fuse the codes to obtain their representations in the unified latent space. To enforce the learning of such a structured latent space, the proposed method adopts a series of constraints including structured ranking regularization, latent code swapping constraint, and distribution supervision on the related partial point clouds. By establishing such a unified and structured latent space, better partial-complete geometry consistency and shape completion accuracy can be achieved. Extensive experiments show that our proposed method consistently outperforms state-of-the-art unsupervised methods on both synthetic ShapeNet and real-world KITTI, ScanNet, and Matterport3D datasets.