  The symmetric product of vector fields on a manifold arises when one studies
the controllability of certain classes of mechanical control systems. A
geometric description of the symmetric product is provided using parallel
transport, along the lines of the flow interpretation of the Lie bracket. This
geometric interpretation of the symmetric product is used to provide an
intrinsic proof of the fact that the distributions closed under the symmetric
product are exactly those distributions invariant under the geodesic flow.
