With the rise of pretrained models, fine-tuning has become increasingly important. However, naive fine-tuning often does not eliminate a model's sensitivity to spurious cues. To understand and address this limitation, we study the geometry of neural network loss landscapes through the lens of mode-connectivity. We tackle two questions: 1) Are models trained on different distributions mode-connected? 2) Can we fine tune a pre-trained model to switch modes? We define a notion of mechanistic similarity based on shared invariances and show linearly-connected modes are mechanistically similar. We find naive fine-tuning yields linearly connected solutions and hence is unable to induce relevant invariances. We also propose and validate a method of "mechanistic fine-tuning" based on our gained insights.