  We construct natural operators connecting the cohomology of the moduli spaces
of stable Higgs bundles with different ranks and genera which, after numerical
specialization, recover the topological mirror symmetry conjecture of
Hausel-Thaddeus concerning $\mathrm{SL}_n$- and $\mathrm{PGL}_n$-Higgs bundles.
This provides a complete description of the cohomology of the moduli space of
stable $\mathrm{SL}_n$-Higgs bundles in terms of the tautological classes, and
gives a new proof of the Hausel-Thaddeus conjecture, proven recently by
Gr\"ochenig-Wyss-Ziegler via p-adic integration.
  Our method is to relate the decomposition theorem for the Hitchin fibration,
using vanishing cycle functors, to the decomposition theorem for the twisted
Hitchin fibration whose supports are simpler.
