  We slightly extend the notion of a natural fibre bundle by requiring
diffeomorphisms of the base to lift to automorphisms of the bundle only
infinitesimally, i.e. at the level of the Lie algebra of vector fields. Spin
structures are natural only in this extended sense. We classify fibre bundles
with this property, assuming a finite dimensional structure group. This
includes all spin structures, but only some generalised spin structures. This
classification links the gauge group G to the topology of space-time.
