We present an approach to learn dense, continuous 2D-3D correspondence distributions over the surface of objects from data with no prior knowledge of visual ambiguities like symmetry. We also present a new method for 6D pose estimation of rigid objects using the learnt distributions to sample, score and refine pose hypotheses. The correspondence distributions are learnt with a contrastive loss, represented in object-specific latent spaces by an encoder-decoder query model and a small fully connected key model. Our method is unsupervised with respect to visual ambiguities, yet we show that the query- and key models learn to represent accurate multi-modal surface distributions. Our pose estimation method improves the state-of-the-art significantly on the comprehensive BOP Challenge, trained purely on synthetic data, even compared with methods trained on real data. The project site is at https://surfemb.github.io/ .