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We carefully analyze the conditions for an abelian gauged linear sigma-modelto exhibit nontrivial IR behavior described by a nonsingular superconformalfield theory determining a superstring vacuum. This is done without referenceto a geometric phase, by associating singular behavior to a noncompact space of(semi-)classical vacua. We find that models determined by reflexivecombinatorial data are nonsingular for generic values of their parameters. Thiscondition has the pleasant feature that the mirror of a nonsingular gaugedlinear sigma-model is another such model, but it is clearly too strong and weprovide an example of a non-reflexive mirror pair. We discuss a weakercondition inspired by considering extremal transitions, which is also mirrorsymmetric and which we conjecture to be sufficient. We apply these ideas toextremal transitions and to understanding the way in which both Berglund-Hubschmirror symmetry and the Vafa-Witten mirror orbifold with discrete torsion canbe seen as special cases of the general combinatorial duality of GLSMs. In theformer case we encounter an example showing that our weaker condition is stillnot necessary.

General Mirror Pairs for Gauged Linear Sigma Models