Title: Steerable Equivariant Representation Learning. Abstract: Pre-trained deep image representations are useful for post-training tasks such as classification through transfer learning, image retrieval, and object detection. Data augmentations are a crucial aspect of pre-training robust representations in both supervised and self-supervised settings. Data augmentations explicitly or implicitly promote \emph{invariance} in the embedding space to the input image transformations. This invariance reduces generalization to those downstream tasks which rely on sensitivity to these particular data augmentations. In this paper, we propose a method of learning representations that are instead \emph{equivariant} to data augmentations. We achieve this equivariance through the use of \emph{steerable} representations. Our representations can be manipulated directly in embedding space via learned linear maps. We demonstrate that our resulting steerable and equivariant representations lead to better performance on transfer learning and robustness: e.g. we improve linear probe top-1 accuracy by between 1\% to 3\% for transfer; and ImageNet-C accuracy by upto 3.4\%. We further show that the steerability of our representations provides significant speedup (nearly $50\times$) for test-time augmentations; by applying a large number of augmentations for out-of-distribution detection, we significantly improve OOD AUC on the ImageNet-C dataset over an invariant representation.