  We revisit partition functions of closed strings on toroidal backgrounds,
including their $\mathbb{Z}_N$ shift orbifolds in the formalism where the
dimension of the target space is doubled to make T-duality manifest. In such a
T-duality covariant formalism, the constraint equation imposes a form of chiral
factorization. Our computation furnishes a non-trivial consistency check for
the quantum worldsheet theory of the doubled sigma model, when strings are
placed on general toroidal backgrounds. The topological term that mixes the
physical space and its T-dual is crucial in demonstrating that chiral
factorization works, and that we obtain the correct partition function after
imposing the constraints. Finally, we discuss how our results extend to
$\mathcal{N}=1$ worldsheet supersymmetry and string worldsheets of higher
genus.
