  Toroidal 3-orbifolds $(S^1)^6/G$, for $G$ a finite group, were some of the
earliest examples of Calabi-Yau 3-orbifolds to be studied in string theory.
While much mathematical progress towards the predictions of string theory has
been made in the meantime, most of it has dealt with hypersurfaces in toric
varieties. As a result, very little is known about curve-counting theories on
toroidal orbifolds. In this paper, we initiate a program to study mirror
symmetry and the Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for toroidal
orbifolds. We focus on the simplest example $[E^3/\mu_3],$ where
$E\subseteq\mathbb{P}^2$ is the elliptic curve $\mathbb{V}(x_0^3+x_1^3+x_2^3).$
We study this orbifold from the point of GIT wall-crossing using the gauged
linear sigma model, a collection of moduli spaces generalizing spaces of stable
maps. Our main result is a mirror symmetry theorem that applies simultaneously
to the different GIT chambers. Using this, we analyze wall-crossing behavior to
obtain an LG/CY correspondence relating the genus-zero Gromov-Witten invariants
of $[E^3/\mu_3]$ to generalized Fan-Jarvis-Ruan-Witten invariants.
