We show the emergence of Clifford algebras of nonsymmetric bilinear forms as cotangent algebras of Kaluza–Klein (KK) spaces pertaining to teleparallel space–times. These spaces are canonically determined by the horizontal differential invariants of Finsler bundles of the type, B′(M)→S(M), where B′(M) is the set of all the tangent frames to a differentiable manifold M, and where S(M) is the sphere bundle. If M is space–time itself, M4, the “geometric phase space,” S(M4), has dimension seven. This reformulation of the horizontal invariants as pertaining to a KK space removes the mismatch between the dimensionality of the tangent frames to M4 and the dimensionality of S(M4). In the KK space, a symmetric tangent metric induces a cotangent metric which is not symmetric in general. An interior covariant derivative in the sense of Kaehler is defined. It involves the antisymmetric part of the cotangent metric, which thus enters electrodynamics and the Dirac equation.