  We present in the context of supersymmetric gauge theories an extension of
the Weyl integration formula, first discovered by Robert Wendt, which applies
to a class of non-connected Lie groups. This allows to count in a systematic
way gauge-invariant chiral operators for these non-connected gauge groups.
Applying this technique to $\mathrm{O}(n)$, we obtain, via the ADHM
construction, the Hilbert series for certain instanton moduli spaces. We
validate our general method and check our results via a Coulomb branch
computation, using three-dimensional mirror symmetry.
