This paper investigates the problem of inertial navigation system (INS) filter design through the lens of symmetry. The extended Kalman filter (EKF) and its variants have been the staple of INS filtering for 50 years. However, recent advances in inertial navigation systems have exploited matrix Lie group structure to design stochastic filters and state observers that have been shown to display superior performance compared to classical solutions. In this work, we explore various symmetries of inertial navigation system, including two novel symmetries that have not been considered in the prior literature, and provide a discussion of the relative strengths and weaknesses of these symmetries in the context of filter design. We show that all the modern variants of the EKF for inertial navigation can be interpreted as the recently proposed equivariant filter (EqF) design methodology applied to different choices of symmetry group for the INS problem. As a direct application of the symmetries presented, we address the filter design problem for a vehicle equipped with an inertial measurement unit (IMU) and a global navigation satellite system (GNSS) receiver, providing a comparative analysis of different modern filter solutions. We believe the collection of symmetries that we present here capture all the sensible choices of symmetry for this problem, and that the analysis provided is indicative of the relative real-world performance potential of the different algorithms for trajectories ensuring full state observability.