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Contact inhibition of locomotion (CIL), in which cells repolarize and move away from contact, is now established as a fundamental driving force in development, repair, and disease biology. Much of what we know of CIL stems from studies on two-dimensional (2D) substrates that do not provide an essential biophysical cue—the curvature of extracellular matrix fibers. We discover rules controlling outcomes of cell–cell collisions on suspended nanofibers and show them to be profoundly different from the stereotyped CIL behavior on 2D substrates. Two approaching cells attached to a single fiber do not repolarize upon contact but rather usually migrate past one another. Fiber geometry modulates this behavior; when cells attach to two fibers, reducing their freedom to reorient, only one cell repolarizes on contact, leading to the cell pair migrating as a single unit. CIL outcomes also change when one cell has recently divided and moves with high speed—cells more frequently walk past each other. Our computational model of CIL in fiber geometries reproduces the core qualitative results of the experiments robustly to model parameters. Our model shows that the increased speed of postdivision cells may be sufficient to explain their increased walk-past rate. We also identify cell–cell adhesion as a key mediator of collision outcomes. Our results suggest that characterizing cell–cell interactions on flat substrates, channels, or micropatterns is not sufficient to predict interactions in a matrix—the geometry of the fiber can generate entirely new behaviors.