Geomagnetically induced currents (GICs) are a major space weather hazard during geomagnetic storms and pose ongoing threats to technologies using long conductor systems. Accurate forecasting of geomagnetic perturbations is essential for estimating GIC risk. Thanks to decades of magnetometer and solar wind observations from the SuperMAG collaboration and upstream space instruments at L1, data-driven modeling of global geomagnetic perturbations has become increasingly feasible. Such models are promising for operational use because they typically run at lower computational cost and can offer higher predictive skill than first-principles, physics-based simulations when observations are abundant. However, this potential has not been fully realized as sparse observational coverage in the Southern Hemisphere (SH) has long limited the generalizability of data-driven models. In this work, we leverage mirror symmetry in ionospheric electrodynamics to improve SH predictions using information from the densely observed Northern Hemisphere (NH). Building on our previous GeoDGP model, we introduce a novel kernel that encodes mirror symmetry into the prior of the Bayesian model, enabling data-driven learning of covariance between interhemispheric conjugate locations. We also experiment with an alternative approach that imposes symmetry as a hard constraint via data augmentation. We evaluate the models on 22 geomagnetic storms and present a case study of the 2024-05-10 Gannon extreme storm. The results show that both methods significantly improve SH predictions, while the kernel-based approach achieves the best performance and maintains NH accuracy comparable to GeoDGP.