Polar electrodynamics is largely controlled by solar wind and magnetospheric forcing. Different conditions can make plasma convection and magnetic field disturbances asymmetric between hemispheres. So far, these asymmetries have been studied in isolation. We present an explanation of how they are linked via displacements of magnetic field line footpoints between hemispheres, under the assumption of ideal magnetohydrodynamics. This displacement has so far been studied only on a point by point basis; here we generalize the concept to a 2D displacement vector field. We estimate displacement fields from average patterns of ionospheric convection using the Weimer et al. (J. Geophys. Res., 2005a, 110, A05306) model. These estimates confirm that the influence of the interplanetary magnetic field extends deep into the magnetosphere, as predicted by models and in-situ observations. Contrary to predictions, the displacement associated with dipole tilt appears uniform across the nightside, and it exceeds the effect of IMF By. While more research is needed to confirm these specific findings, our results demonstrate how ionospheric observations can be used to infer magnetospheric morphology, and that the displacement field is a critical component for understanding geospace as a coupled two-hemisphere system.