<p>We show that a family of birational transformations that relate toric Fano 3-folds defined by reflexive lattice polytopes can be identified with mass deformations of corresponding 2<i>d</i> (0, 2) supersymmetric quiver gauge theories. These theories are realized by a Type IIA brane configuration known as brane brick models. We further show that the same family of birational transformations extends to more general toric Calabi-Yau 4-folds, including those defined by non-reflexive toric diagrams. Under these birational transformations, the mesonic moduli spaces of the associated abelian 2<i>d</i> (0, 2) supersymmetric gauge theories and brane brick models share the same number of generators and the same Hilbert series when refined only under the U(1)_<i>R</i> symmetry. Since these transformations categorize toric Calabi-Yau 4-folds and their corresponding 2<i>d</i> (0, 2) supersymmetric gauge theories into non-trivial equivalence classes, we anticipate that our findings will pave the way for a ‘Minimal Model Program’ for quiver gauge theories corresponding to toric Calabi-Yau manifolds.</p>