A generalized Clifford manifold is proposed in which there are coordinates
not only for the basis vector generators, but for each element of the Clifford
group, including the identity scalar. These new quantities are physically
interpreted to represent internal structure of matter (e.g. classical or
quantum spin). The generalized Dirac operator must now include differentiation
with respect to these higher order geometric coordinates. In a Riemann space,
where the magnitude and rank of geometric objects are preserved under
displacement, these new terms modify the geodesics. One possible physical
interpretation is natural coupling of the classical spin to linear motion,
providing a new derivation of the Papapetrou equations. A generalized curvature
is proposed for the Clifford manifold in which the connection does not preserve
the rank of a multivector under parallel transport, e.g. a vector may be
``rotated'' into a scalar.