Grid cells in the mammalian brain are fundamental to spatial navigation, \emph{i.e.}, how animals perceive and interact with their environment. Traditionally, grid cells are thought to encode the physical position of an animal. However, recent findings show that their firing patterns become distorted in the presence of significant spatial landmarks such as rewarded locations. This introduces a novel perspective of dynamic, subjective, and action-relevant interactions between grid cells and environmental cues. Here, we propose a practical and theoretical framework to quantify and explain these interactions. To this end, we train path-integrating recurrent neural networks (piRNNs) on a spatial navigation task, whose goal is to predict the agent's position with a special focus on rewarded locations. Grid-like neurons naturally emerge from the training of piRNNs, which allows us to investigate how the two aspects of the task, space and reward, are integrated in their firing patterns. We find that geometry, but not topology, of the grid cell population code becomes distorted. Surprisingly, these distortions are global in the firing patterns of the grid cells despite local changes in the reward. Our results indicate that the preserved hexagonal firing rates after fine-tuning may retain the spatial navigation abilities, whereas the global distortions emerging after training for location-specific reward information may encode dynamically changing environmental cues. By bridging the gap between computational models and biological reality of spatial navigation under reward information, we offer new insights into how neural systems prioritize environmental landmarks in their spatial navigation code.