In Abłamowicz and Fauser [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras I: The transposition map, Linear Multilinear Alg. (to appear)] a signature ϵ = (p, q)-dependent transposition anti-involution of real Clifford algebras Cℓ p,q for non-degenerate quadratic forms was introduced. In Abłamowicz and Fauser [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Alg. (to appear)] we showed that, depending on the value of (p − q) mod 8, the map 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 [R. Abłamowicz and B. Fauser, Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple, Comput. Phys. Commun. 170 (2005), pp. 115–130]. We provide a full signature (p, q)-dependent classification of the invariance groups of this product for p + q ≤ 9. The map is identified as the ‘star’ map known [D.S. Passman, The Algebraic Structure of Group Rings, Robert E. Krieger Publishing Company, Malabar, Florida, 1985] from the theory of (twisted) group algebras where the Clifford algebra Cℓ p,q is seen as a twisted group ring n = p + q. We discuss important subgroups of a stabilizer group G p,q (f) of a primitive idempotent f and we relate their transversals to spinor bases in spinor spaces realized as minimal left ideals Cℓ p,q f.