  We prove a general mirror duality theorem for a subalgebra $U$ of a simple
conformal vertex algebra $A$ and its commutant $V=\mathrm{Com}_A(U)$.
Specifically, we assume that $A\cong\bigoplus_{i\in I} U_i\otimes V_i$ as a
$U\otimes V$-module, where the $U$-modules $U_i$ are simple and distinct and
are objects of a semisimple braided ribbon category of $U$-modules, and the
$V$-modules $V_i$ are semisimple and contained in a (not necessarily rigid)
braided tensor category of $V$-modules. We also assume $U=\mathrm{Com}_A(V)$.
Under these conditions, we construct a braid-reversed tensor equivalence $\tau:
\mathcal{U}_A\rightarrow\mathcal{V}_A$, where $\mathcal{U}_A$ is the semisimple
category of $U$-modules with simple objects $U_i$, $i\in I$, and
$\mathcal{V}_A$ is the category of $V$-modules whose objects are finite direct
sums of the $V_i$. In particular, the $V$-modules $V_i$ are simple and
distinct, and $\mathcal{V}_A$ is a rigid tensor category. As an application, we
find a rigid semisimple tensor subcategory of modules for the Virasoro algebra
at central charge $13+6p+6p^{-1}$, $p\in\mathbb{Z}_{\geq 2}$, which is braided
tensor equivalent to an abelian $3$-cocycle twist of the category of
finite-dimensional $\mathfrak{sl}_2$-modules. Consequently, the Virasoro vertex
operator algebra at central charge $13+6p+6p^{-1}$ is the
$PSL_2(\mathbb{C})$-fixed-point subalgebra of a simple conformal vertex algebra
$\mathcal{W}(-p)$, analogous to the realization of the Virasoro vertex operator
algebra at central charge $13-6p-6p^{-1}$ as the
$PSL_2(\mathbb{C})$-fixed-point subalgebra of the triplet algebra
$\mathcal{W}(p)$.
