The entangling power of a unitary operator acting on a bipartite Hilbert space measures the entanglement it generates from product states, averaged over the inputs. A finite-dimensional unitary has a spectral decomposition $U=\sum_{a=1}^{n}e^{i\theta_
The entangling power of a unitary operator acting on a bipartite Hilbert space measures the entanglement it generates from product states, averaged over the inputs. A finite-dimensional unitary has a spectral decomposition U=\sum_{a=1}^{n}e^{i\theta_a}P_a, where e^{i\theta_a} are the eigenvalues, P_a the corresponding eigen-projectors, and n is the number of distinct eigenvalues. After removing an overall phase, the entangling power is a function on the (n-1)-torus of relative eigenphases at fixed spectral projectors. We prove that this function is stationary at all 2^{n-1} points on the torus where every relative phase is 0 or \pi, which we define as corners. Up to an overall phase, U at each corner is a generalized reflection R=I-2Q satisfying R^2=I, where Q is the sum of spectral projectors whose relative phase is \pi. At the corner the entangling power is expressed in terms of seven local-unitary invariants of Q. A unitary gate U can be realized as a corner of some projector family if and only if U^2\proptoI, a condition satisfied by many Clifford and non-Clifford gates. We illustrate the theorem with two-qubit gates, SU(N) channel decompositions, and two-site spin chains, obtaining examples of minima, maxima, and saddle points. In addition, a corner that is a saddle point on the full phase torus can appear as a local maximum or minimum along different time-evolution trajectories.