  It is shown that every bundle $\varSigma\to M$ of complex spinor modules over
the Clifford bundle $\Cl(g)$ of a Riemannian space $(M,g)$ with local model
$(V,h)$ is associated with an lpin ("Lipschitz") structure on $M$, this being a
reduction of the ${\Ort}(h)$-bundle of all orthonormal frames on M to the
Lipschitz group $\Lpin(h)$ of all automorphisms of a suitably defined spin
space. An explicit construction is given of the total space of the
$\Lpin(h)$-bundle defining such a structure. If the dimension m of M is even,
then the Lipschitz group coincides with the complex Clifford group and the lpin
structure can be reduced to a pin$^{c}$ structure. If m=2n-1, then a spinor
module $\varSigma$ on M is of the Cartan type: its fibres are 2^n-dimensional
and decomposable at every point of M, but the homomorphism of bundles of
algebras $\Cl(g)\to\End\varSigma$ globally decomposes if, and only if, M is
orientable. Examples of such bundles are given. The topological condition for
the existence of an lpin structure on an odd-dimensional Riemannian manifold is
derived and illustrated by the example of a manifold admitting such a
structure, but no pin^c structure.
