Transformation-based methods for shape analysis offer a consistent framework to model the geometrical content of images. Most often relying on diffeomorphic transforms, they lack however the ability to properly handle texture and differing topological content. Conversely, modern deep learning methods offer a very efficient way to analyze image textures. Building on the theory of metamorphoses, which models images as combined intensity-domain and spatial-domain transforms of a prototype, we introduce the "metamorphic" auto-encoding architecture. This class of neural networks is interpreted as a Bayesian generative and hierarchical model, allowing the joint estimation of the network parameters, a representative prototype of the training images, as well as the relative importance between the geometrical and texture contents. We give arguments for the practical relevance of the learned prototype and Euclidean latent-space metric, achieved thanks to an explicit normalization layer. Finally, the ability of the proposed architecture to learn joint and relevant shape and appearance representations from image collections is illustrated on BraTs 2018 datasets, showing in particular an encouraging step towards personalized numerical simulation of tumors with data-driven models.