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Image hash codes are produced by binarizing the embeddings of convolutional neural networks (CNN) trained for either classification or retrieval. While proxy embeddings achieve good performance on both tasks, they are non-trivial to binarize, due to a rotational ambiguity that encourages non-binary embeddings. The use of a fixed set of proxies (weights of the CNN classification layer) is proposed to eliminate this ambiguity, and a procedure to design proxy sets that are nearly optimal for both classification and hashing is introduced. The resulting hash-consistent large margin (HCLM) proxies are shown to encourage saturation of hashing units, thus guaranteeing a small binarization error, while producing highly discriminative hash-codes. A semantic extension (sHCLM), aimed to improve hashing performance in a transfer scenario, is also proposed. Extensive experiments show that sHCLM embeddings achieve significant improvements over state-of-the-art hashing procedures on several small and large datasets, both within and beyond the set of training classes.