Humans effortlessly grasp the connection between sketches and real-world objects, even when these sketches are far from realistic. Moreover, human sketch understanding goes beyond categorization -- critically, it also entails understanding how individual elements within a sketch correspond to parts of the physical world it represents. What are the computational ingredients needed to support this ability? Towards answering this question, we make two contributions: first, we introduce a new sketch-photo correspondence benchmark, PSC6k, containing 150K annotations of 6250 sketch-photo pairs across 125 object categories, augmenting the existing Sketchy dataset with fine-grained correspondence metadata. Second, we propose a self-supervised method for learning dense correspondences between sketch-photo pairs, building upon recent advances in correspondence learning for pairs of photos. Our model uses a spatial transformer network to estimate the warp flow between latent representations of a sketch and photo extracted by a contrastive learning-based ConvNet backbone. We found that this approach outperformed several strong baselines and produced predictions that were quantitatively consistent with other warp-based methods. However, our benchmark also revealed systematic differences between predictions of the suite of models we tested and those of humans. Taken together, our work suggests a promising path towards developing artificial systems that achieve more human-like understanding of visual images at different levels of abstraction. Project page: https://photo-sketch-correspondence.github.io