A popular explanation for the microscopic origin of Bekenstein's black hole entropy is the conjecture that this entropy can be ascribed to a collection of (1+1) dimensional conformal field theories that reside at the horizon, and defined on the two-plane perpendicular to the horizon. If this is to be the case, then the Einstein equations must force the Ricci curvature to possess a high degree of symmetry at the horizon. We test this hypothesis by working directly with the spacetime geometry for a generic rotating black hole --- constrained only by the existence of a stationary non-static Killing horizon, and with otherwise arbitrary matter content --- to show that the Einstein tensor block diagonalizes on the horizon. This is a specific example of an "enhanced symmetry" that manifests only at the horizon itself.