The problem of graph matching under node and pair-wise constraints is fundamental in areas as diverse as combinatorial optimization, machine learning or computer vision, where representing both the relations between nodes and their neighborhood structure is essential. We present an end-to-end model that makes it possible to learn all parameters of the graph matching process, including the unary and pairwise node neighborhoods, represented as deep feature extraction hierarchies. The challenge is in the formulation of the different matrix computation layers of the model in a way that enables the consistent, efficient propagation of gradients in the complete pipeline from the loss function, through the combinatorial optimization layer solving the matching problem, and the feature extraction hierarchy. Our computer vision experiments and ablation studies on challenging datasets like PASCAL VOC keypoints, Sintel and CUB show that matching models refined end-to-end are superior to counterparts based on feature hierarchies trained for other problems.