  Odd-dimensional Riemannian spaces that are non-orientable, but have a pin
structure, require the consideration of the twisted adjoint representation of
the corresponding pin group. It is shown here how the Dirac operator should be
modified, also on even-dimensional spaces, to make it equivariant with respect
to the action of that group when the twisted adjoint representation is used in
the definition of the pin structure. An explicit description of a pin structure
on a hypersurface, defined by its immersion in a Euclidean space, is used to
derive a "Schroedinger" transform of the Dirac operator in that case. This is
then applied to obtain - in a simple manner - the spectrum and eigenfunctions
of the Dirac operator on spheres and real projective spaces.
