Identifying the type of kidney stones can allow urologists to determine their cause of formation, improving the prescription of appropriate treatments to diminish future relapses. Currently, the associated ex-vivo diagnosis (known as Morpho-constitutional Analysis, MCA) is time-consuming, expensive and requires a great deal of experience, as it requires a visual analysis component that is highly operator dependant. Recently, machine learning methods have been developed for in-vivo endoscopic stone recognition. Deep Learning (DL) based methods outperform non-DL methods in terms of accuracy but lack explainability. Despite this trade-off, when it comes to making high-stakes decisions, it's important to prioritize understandable Computer-Aided Diagnosis (CADx) that suggests a course of action based on reasonable evidence, rather than a model prescribing a course of action. In this proposal, we learn Prototypical Parts (PPs) per kidney stone subtype, which are used by the DL model to generate an output classification. Using PPs in the classification task enables case-based reasoning explanations for such output, thus making the model interpretable. In addition, we modify global visual characteristics to describe their relevance to the PPs and the sensitivity of our model's performance. With this, we provide explanations with additional information at the sample, class and model levels in contrast to previous works. Although our implementation's average accuracy is lower than state-of-the-art (SOTA) non-interpretable DL models by 1.5 %, our models perform 2.8% better on perturbed images with a lower standard deviation, without adversarial training. Thus, Learning PPs has the potential to create more robust DL models.