  Let X and X' be compact Riemann surfaces of genus at least three. Let G and
G' be nontrivial connected semisimple linear algebraic groups over C. If some
components $M_{DH}^d(X,G)$ and $M_{DH}^{d'}(X',G')$ of the associated
Deligne--Hitchin moduli spaces are biholomorphic, then X' is isomorphic to X or
to the conjugate Riemann surface $\bar{X}$.
