  Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of
finite-length modules for the Virasoro Lie algebra at central charge $25$ whose
composition factors are the irreducible quotients of reducible Verma modules.
We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a
semisimple tensor subcategory that is braided tensor equivalent to an abelian
$3$-cocycle twist of the category of finite-dimensional
$\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type
subcategory is braid-reversed tensor equivalent to a similar category for the
Virasoro algebra at central charge $1$. As an application, we construct a
simple conformal vertex algebra which contains the Virasoro vertex operator
algebra of central charge $25$ as a $PSL_2(\mathbb{C})$-orbifold. We also use
our results to study Arakawa's chiral universal centralizer algebra of $SL_2$
at level $-1$, showing that it has a symmetric tensor category of
representations equivalent to $\mathrm{Rep}\,PSL_2(\mathbb{C})$. This algebra
is an extension of the tensor product of Virasoro vertex operator algebras of
central charges $1$ and $25$, analogous to the modified regular representations
of the Virasoro algebra constructed earlier for generic central charges by I.
Frenkel-Styrkas and I. Frenkel-M. Zhu.
