For the complex representations of the real Clifford algebras, the associated reality structure on the complex Clifford algebra descends to the space of spinors (via the explicit construction in terms of minimal ideals, for instance). In this way, we obtain the complex conjugate Δ of the representation Δ, and the following isomorphism is seen to hold: Δ Δ ¯ ¯ ≅ ≅ σ σ − − Δ Δ ∗ ∗