We propose C3DPO, a method for extracting 3D models of deformable objects from 2D keypoint annotations in unconstrained images. We do so by learning a deep network that reconstructs a 3D object from a single view at a time, accounting for partial occlusions, and explicitly factoring the effects of viewpoint changes and object deformations. In order to achieve this factorization, we introduce a novel regularization technique. We first show that the factorization is successful if, and only if, there exists a certain canonicalization function of the reconstructed shapes. Then, we learn the canonicalization function together with the reconstruction one, which constrains the result to be consistent. We demonstrate state-of-the-art reconstruction results for methods that do not use ground-truth 3D supervision for a number of benchmarks, including Up3D and PASCAL3D+.