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Given a collection of images, humans are able to discover landmarks by modeling the shared geometric structure across instances. This idea of geometric equivariance has been widely used for the unsupervised discovery of object landmark representations. In this paper, we develop a simple and effective approach by combining instance-discriminative and spatially-discriminative contrastive learning. We show that when a deep network is trained to be invariant to geometric and photometric transformations, representations emerge from its intermediate layers that are highly predictive of object landmarks. Stacking these across layers in a "hypercolumn" and projecting them using spatially-contrastive learning further improves their performance on matching and few-shot landmark regression tasks. We also present a unified view of existing equivariant and invariant representation learning approaches through the lens of contrastive learning, shedding light on the nature of invariances learned. Experiments on standard benchmarks for landmark learning, as well as a new challenging one we propose, show that the proposed approach surpasses prior state-of-the-art.