In this paper, we present ShapeMatcher, a unified self-supervised learning framework for joint shape canonicalization, segmentation, retrieval and deformation. Given a partially-observed object in an arbitrary pose, we first canonicalize the object by extracting point-wise affine-invariant features, disentangling inherent structure of the object with its pose and size. These learned features are then leveraged to predict semantically consistent part segmentation and corresponding part centers. Next, our lightweight retrieval module aggregates the features within each part as its retrieval token and compare all the tokens with source shapes from a pre-established database to identify the most geometrically similar shape. Finally, we deform the retrieved shape in the deformation module to tightly fit the input object by harnessing part center guided neural cage deformation. The key insight of ShapeMaker is the simultaneous training of the four highly-associated processes: canonicalization, segmentation, retrieval, and deformation, leveraging cross-task consistency losses for mutual supervision. Extensive experiments on synthetic datasets PartNet, ComplementMe, and real-world dataset Scan2CAD demonstrate that ShapeMaker surpasses competitors by a large margin.