Unsupervised domain adaptation aims to transfer and adapt knowledge learned from a labeled source domain to an unlabeled target domain. Key components of unsupervised domain adaptation include: (a) maximizing performance on the source, and (b) aligning the source and target domains. Traditionally, these tasks have either been considered as separate, or assumed to be implicitly addressed together with high-capacity feature extractors. In this paper, we advance a third broad approach; which we term SALT. The core idea is to consider alignment as an auxiliary task to the primary task of maximizing performance on the source. The auxiliary task is made rather simple by assuming a tractable data geometry in the form of subspaces. We synergistically allow certain parameters derived from the closed-form auxiliary solution, to be affected by gradients from the primary task. The proposed approach represents a unique fusion of geometric and model-based alignment with gradient-flows from a data-driven primary task. SALT is simple, rooted in theory, and outperforms state-of-the-art on multiple standard benchmarks.