The last years have seen a growing body of literature on data-driven pedestrian inertial navigation. However, despite this, it is still unclear how to efficiently combine classical models and other a priori information with existing machine learning frameworks. In this paper, we first categorize existing approaches to data-driven pedestrian inertial navigation, including approaches where a machine learning algorithm is embedded into an overarching classical framework and purely data-driven frameworks. We then propose an estimation framework where navigation estimates obtained by classical means are fed to a machine learning algorithm which is trained to correct and improve the estimates. Further, we describe three symmetries that can be used to constrain the proposed estimation framework and thereby improve its performance. These are 1) the rotational symmetry of pedestrian dynamics, 2) the rotational symmetry of the sensors, and 3) the temporal symmetry of pedestrian dynamics. To demonstrate the usefulness of the proposed framework, we use data from foot-mounted inertial sensors utilizing zero-velocity updates under mixed walking and running. Machine learning corrections are implemented using both neural networks and Gaussian processes.