Graph Neural Networks (GNNs) have been shown to achieve competitive results to tackle graph-related tasks, such as node and graph classification, link prediction and node and graph clustering in a variety of domains. Most GNNs use a message passing framework and hence are called MPNNs. Despite their promising results, MPNNs have been reported to suffer from over-smoothing, over-squashing and under-reaching. Graph rewiring and graph pooling have been proposed in the literature as solutions to address these limitations. However, most state-of-the-art graph rewiring methods fail to preserve the global topology of the graph, are not differentiable (inductive) and require the tuning of hyper-parameters. In this paper, we propose DiffWire, a novel framework for graph rewiring in MPNNs that is principled, fully differentiable and parameter-free by leveraging the Lovász bound. Our approach provides a unified theory for graph rewiring by proposing two new, complementary layers in MPNNs: first, CTLayer, a layer that learns the commute times and uses them as a relevance function for edge re-weighting; second, GAPLayer, a layer to optimize the spectral gap, depending on the nature of the network and the task at hand. We empirically validate the value of our proposed approach and each of these layers separately with benchmark datasets for graph classification. DiffWire brings together the learnability of commute times to related definitions of curvature, opening the door to the development of more expressive MPNNs.