We present 3DPointCaps++ for learning robust, flexible and generalizable 3D object representations without requiring heavy annotation efforts or supervision. Unlike conventional 3D generative models, our algorithm aims for building a structured latent space where certain factors of shape variations, such as object parts, can be disentangled into independent sub-spaces. Our novel decoder then acts on these individual latent sub-spaces (i.e. capsules) using deconvolution operators to reconstruct 3D points in a self-supervised manner. We further introduce a cluster loss ensuring that the points reconstructed by a single capsule remain local and do not spread across the object uncontrollably. These contributions allow our network to tackle the challenging tasks of part segmentation, part interpolation/replacement as well as correspondence estimation across rigid / non-rigid shape, and across / within category. Our extensive evaluations on ShapeNet objects and human scans demonstrate that our network can learn generic representations that are robust and useful in many applications.