Deformable image registration poses a challenging problem where, unlike most deep learning tasks, a complex relationship between multiple coordinate systems has to be considered. Although data-driven methods have shown promising capabilities to model complex non-linear transformations, existing works employ standard deep learning architectures assuming they are general black-box solvers. We argue that understanding how learned operations perform pattern-matching between the features in the source and target domains is the key to building robust, data-efficient, and interpretable architectures. We present a theoretical foundation for designing an interpretable registration framework: separated feature extraction and deformation modeling, dynamic receptive fields, and a data-driven deformation functions awareness of the relationship between both spatial domains. Based on this foundation, we formulate an end-to-end process that refines transformations in a coarse-to-fine fashion. Our architecture employs spatially continuous deformation modeling functions that use geometric deep-learning principles, therefore avoiding the problematic approach of resampling to a regular grid between successive refinements of the transformation. We perform a qualitative investigation to highlight interesting interpretability properties of our architecture. We conclude by showing significant improvement in performance metrics over state-of-the-art approaches for both mono- and multi-modal inter-subject brain registration, as well as the challenging task of longitudinal retinal intra-subject registration. We make our code publicly available