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A signature epsilon=(p,q) dependent transposition anti-involution T of realClifford algebras Cl_{p,q} for non-degenerate quadratic forms was introduced in[arXiv.1005.3554v1]. In [arXiv.1005.3558v1] we showed that, depending on thevalue of (p-q) mod 8, the map T gives rise to transposition, complex Hermitian,or quaternionic Hermitian conjugation of representation matrices in spinorrepresentation. The resulting scalar product is in general different from thetwo known standard scalar products [Lounesto, Clifford algebras and Spinors2001]. We provide a full signature (p,q) dependent classification of theinvariance groups Gpq_{p,q} of this product for p+q <= 9. The map T isidentified as the "star" map known [Passmann, The Algebraic Structure of GroupRings 1985] from the theory of (twisted) group algebras, where the Cliffordalgebra Cl_{p,q} is seen as a twisted group ring k^t[(Z_2)^n], n=p+q. Wediscuss and list important subgroups of stabilizer groups Gpq(f)_{p,q} andtheir transversals in relation to generators of spinor spaces.

On the Transposition Anti-Involution in Real Clifford Algebras III: The Automorphism Group of the Transposition Scalar Product on Spinor Spaces