  The central result of this paper is an identification of the shifted Cartier
dual of the moduli stack $\mathcal{M}_{\mathfrak{g}}(C)$ of
$\widetilde{G}$-Higgs bundles on $C$ of arbitrary degree (modulo shifts by
$Z(\widetilde{G})$) with a quotient of the Langlands dual stack
$\mathcal{M}_{^L\mathfrak{g}}(C)$. Via hyperk\"ahler rotation, this may
equivalently be viewed as the identification of an SYZ fibration relating
Hitchin systems for arbitrary Langlands dual semisimple groups, coupled to
nontrivial finite $B$-fields. As a corollary certain self-dual stacks
$\frac{\mathcal{M}_{\mathfrak{g}}(C)}{\Gamma}$ are observed to exist, which I
conjecture to be the Coulomb branches for the 3d reduction of the 4d
$\mathcal{N}=2$ theories of class $\mathcal{S}$.
