Mirror symmetry of Calabi—Yau spaces is best understood for families presented as complete intersections in toric varieties; these models have a description as the low-energy limit of Abelian gauged linear sigma models (GLSMs). We investigate the combinatorial conditions on GLSM data such that the generic member of the family determines a non-singular low energy theory. A sufficient condition is reflexivity; this has the pleasant feature that the mirror of a reflexive model is reflexive. This condition is certainly not necessary, in particular it is not preserved by extremal transitions. We propose a weaker condition that is preserved, but is also too strong. Along the way we describe how the Berglund—Hubsch mirror construction is related to the Batyrev—Boris combinatorial duality related to Abelian duality. We study the locus in parameter space along which the model becomes singular and the invariance of this under mirror symmetry, finding support for the observation of Hori and Vafa that mirror symmetry is most naturally stated in terms of local Calabi—Yau models.