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=∑a=1neiθaPa, where
eiθa are the eigenvalues,
Pa 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
2n−1 points on the torus where every relative phase is
0 or
π, which we define as \textit{corners}. Up to an overall phase,
U at each corner is a generalized reflection
R=I−2Q satisfying
R2=I, where
Q is the sum of spectral projectors whose relative phase is
π. 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
U2∝I, 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.