Smart industrial solutions that can address multiple tasks require models that understand the underlying processes. We examine representation learning for modeling the injection moulding process as an example of such solutions. We consider its data efficiency and multi-modal benefit compared to classical machine learning alternatives. To this end, we synthesize a process-informed dataset based on a real reference cycle of an Erlanger Träger part. We then introduce MouldIt, a temporal convolution and attention based variational autoencoder that reconstructs in-mould signals, while classifying parts as acceptable or as one of our synthetic failure modes. We evaluate MouldIt on two tasks, known-class classification and unknown-class detection, across nine dataset sizes in single-modal and multi-modal settings. We compare it with an accept-all baseline and the classical methods k-nearest neighbors, logistic regression, random forests, and principal component analysis, which we adapt to support both tasks. MouldIt outperforms classical methods across both tasks from 4000 cycles in the single-modal setting and from 2000 cycles in the multi-modal setting. It achieves part sorting losses of 6.0 $\pm$ 0.7% and 1.0 $\pm$ 0.3% in the single-modal and multi-modal settings respectively, ranking first across all baselines and a combination of principal component analysis and random forests.