Deep anomaly detection methods learn representations that separate between normal and anomalous samples. Very effective representations are obtained when powerful externally trained feature extractors (e.g. ResNets pre-trained on ImageNet) are fine-tuned on the training data which consists of normal samples and no anomalies. However, this is a difficult task that can suffer from catastrophic collapse, i.e. it is prone to learning trivial and non-specific features. In this paper, we propose a new loss function which can overcome failure modes of both center-loss and contrastive-loss methods. Furthermore, we combine it with a confidence-invariant angular center loss, which replaces the Euclidean distance used in previous work, that was sensitive to prediction confidence. Our improvements yield a new anomaly detection approach, based on $\textit{Mean-Shifted Contrastive Loss}$, which is both more accurate and less sensitive to catastrophic collapse than previous methods. Our method achieves state-of-the-art anomaly detection performance on multiple benchmarks including $97.5\%$ ROC-AUC on the CIFAR-10 dataset.