In the oriented swap process, N ordered particles perform adjacent swaps at random times until they reach the reverse configuration. The last passage percolation model encodes the maximal time spent travelling along directed lattice paths in a random environment. We present new exact distributional identities connecting these two models. In particular, the absorbing time of the oriented swap process has the same law as the point-to-line last passage percolation. They both converge, under an appropriate scaling limit as the size of the system grows, to the GOE Tracy-Widom distribution from random matrix theory. Three celebrated combinatorial bijections will make an appearance: the RSK, Burge, and Edelman-Greene correspondences.