  A signature epsilon=(p,q) dependent transposition anti-involution T of real
Clifford algebras Cl_{p,q} for non-degenerate quadratic forms was introduced in
[arXiv.1005.3554v1]. In [arXiv.1005.3558v1] we showed that, depending on the
value of (p-q) mod 8, the map T gives rise to transposition, complex Hermitian,
or quaternionic Hermitian conjugation of representation matrices in spinor
representation. The resulting scalar product is in general different from the
two known standard scalar products [Lounesto, Clifford algebras and Spinors
2001]. We provide a full signature (p,q) dependent classification of the
invariance groups Gpq_{p,q} of this product for p+q <= 9. The map T is
identified as the "star" map known [Passmann, The Algebraic Structure of Group
Rings 1985] from the theory of (twisted) group algebras, where the Clifford
algebra Cl_{p,q} is seen as a twisted group ring k^t[(Z_2)^n], n=p+q. We
discuss and list important subgroups of stabilizer groups Gpq(f)_{p,q} and
their transversals in relation to generators of spinor spaces.
