Mechanical cues from the extracellular microenvironment play a central role in regulating the structure, function and fate of living cells. In this work we experimentally and theoretically study one of the most striking manifestations of cellular mechanosensitivity – cell reorientation to a uniform angle in response to physiologically-relevant cyclic stretching. We first show that existing models are incompatible with our extensive measurements of cell reorientation. We then propose a fundamentally new theory, that shows that energy minimization, at the level of the cell-substrate adhesion sites, drives the reorientation process. This theory is in excellent quantitative agreement with the complete temporal reorientation dynamics of individual cells, measured over a wide range of experimental conditions, and offers new venues for predicting and controlling cell behavior in response to microenvironmental mechanical cues.