Present and future sensors are diversifying from traditional quad polarimetric mode of synthetic aperture radar acquisition. Thus, an approach that is interpretative in nature and applicable across polarimetric modes is required. In this context, the geodesic distance (GD)-based approach within the polarimetric synthetic aperture radar (PolSAR) literature is seen as an eigenvalue-decomposition free approach to interpret and analyze quad PolSAR data. This approach is highly adaptive toward applications due to its ability to compare the SAR observation with a known scatterer/model, or with another SAR observation in general providing a means for direct interpretation. In this work, we show that the GD (originally defined for the quad polarization mode) is generalizable across any arbitrary SAR polarimetric mode while retaining its simple form for ready computation. We show that the GD-based approach provides level ground for comparison of different polarimetric modes given a fixed application. We demonstrate it using change detection as the chosen application. In addition, we show how the behavior of the three roll-invariant GD-based parameters change under different polarimetric modes (e.g., quad, dual, and compact polarization modes). We also discuss how the GD-based approach can also be adapted to ground range detected (GRD) product data, which is presently available from Sentinel-1 and widely used in many applications. However, in this case, we show that only one of the three GD-derived parameters can be defined. We believe this work will make the GD-based approach important for present and future PolSAR applications cutting across sensors and its available polarimetric modes.