Electroencephalography (EEG) provides access to neuronal dynamics non-invasively with millisecond resolution, rendering it a viable method in neuroscience and healthcare. However, its utility is limited as current EEG technology does not generalize well across domains (i.e., sessions and subjects) without expensive supervised re-calibration. Contemporary methods cast this transfer learning (TL) problem as a multi-source/-target unsupervised domain adaptation (UDA) problem and address it with deep learning or shallow, Riemannian geometry aware alignment methods. Both directions have, so far, failed to consistently close the performance gap to state-of-the-art domain-specific methods based on tangent space mapping (TSM) on the symmetric positive definite (SPD) manifold. Here, we propose a theory-based machine learning framework that enables, for the first time, learning domain-invariant TSM models in an end-to-end fashion. To achieve this, we propose a new building block for geometric deep learning, which we denote SPD domain-specific momentum batch normalization (SPDDSMBN). A SPDDSMBN layer can transform domain-specific SPD inputs into domain-invariant SPD outputs, and can be readily applied to multi-source/-target and online UDA scenarios. In extensive experiments with 6 diverse EEG brain-computer interface (BCI) datasets, we obtain state-of-the-art performance in inter-session and -subject TL with a simple, intrinsically interpretable network architecture, which we denote TSMNet.