Motivated by considerations on curvature constancy and fat bundles, we introduce the notion of even Clifford structures on Riemannian manifolds, a framework generalizing Kähler and quaternion-Kähler geometries. We give the complete classification of manifolds carrying parallel even Clifford structures: Riemannian products of quaternion-Kähler manifolds, several classes of 8-dimensional manifolds, families of real, complex and quaternionic Grassmannians, as well as Rosenfeld's elliptic projective planes, which are symmetric spaces associated to the exceptional simple Lie groups. As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.