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Adopting Convolutional Neural Networks (CNNs) in the daily routine of primary diagnosis requires not only near-perfect precision, but also a sufficient degree of generalization to data acquisition shifts and transparency. Existing CNN models act as black boxes, not ensuring to the physicians that important diagnostic features are used by the model. Building on top of successfully existing techniques such as multi-task learning, domain adversarial training and concept-based interpretability, this paper addresses the challenge of introducing diagnostic factors in the training objectives. Here we show that our architecture, by learning end-to-end an uncertainty-based weighting combination of multi-task and adversarial losses, is encouraged to focus on pathology features such as density and pleomorphism of nuclei, e.g. variations in size and appearance, while discarding misleading features such as staining differences. Our results on breast lymph node tissue show significantly improved generalization in the detection of tumorous tissue, with best average AUC 0.89 (0.01) against the baseline AUC 0.86 (0.005). By applying the interpretability technique of linearly probing intermediate representations, we also demonstrate that interpretable pathology features such as nuclei density are learned by the proposed CNN architecture, confirming the increased transparency of this model. This result is a starting point towards building interpretable multi-task architectures that are robust to data heterogeneity. Our code is available at https://github.com/maragraziani/multitask_adversarial