  We derive conjectures, called genus 1 Enumerative mirror symmetry for moduli
spaces of Higgs bundles, which relate curve-counting invariants of moduli
spaces of Higgs $\mathrm{SL}_r$-bundles to curve-counting invariants of moduli
spaces of Higgs $\mathrm{PGL}_r$-bundles. This contrasts with Enumerative
mirror symmetry for Calabi-Yau 3-folds which relates curve-counting invariants
to periods. We also provide extensive mathematical evidence for these
conjectures. The conjectures are obtained with the help of the theory of
quasimaps to moduli spaces of sheaves, Tanaka-Thomas's construction of
Vafa-Witten theory, Jiang-Kool's enumerative S-duality of Vafa-Witten
invariants and Manschot-Moore's calculations. We use the latter together with
some basic computations to give a complete list of conjectural expressions for
genus 1 quasimap invariants for all prime ranks. They have many interesting
properties, among which is quantum $\chi$-independence. The wall-crossing to
Gromov-Witten theory is also thoroughly discussed.
