  In this paper we extend the nonabelian mirror proposal of two of the authors
from two-dimensional gauge theories with connected gauge groups to the case of
O(k) gauge groups with discrete theta angles. We check our proposed extension
by counting and comparing vacua in mirrors to known dual two-dimensional
(S)O(k) gauge theories. The mirrors in question are Landau-Ginzburg orbifolds,
and for mirrors to O(k) gauge theories, the critical loci of the mirror
superpotential often intersect fixed-point loci, so that to count vacua, one
must take into account twisted sector contributions. This is a technical
novelty relative to mirrors of gauge theories with connected gauge groups, for
which critical loci do not intersect fixed-point loci and so no orbifold
twisted sector contributions are pertinent. The vacuum computations turn out to
be a rather intricate test of the proposed mirrors, in particular as untwisted
sector states in the mirror to one theory are often exchanged with twisted
sector states in the mirror to the dual. In cases with nontrivial IR limits, we
also check that central charges computed from the Landau-Ginzburg mirrors match
those expected for the IR SCFTs.
