  This paper aims to shed light on what becomes of discrete torsion within
heterotic orbifolds when they are resolved to smooth geometries. Gauged Linear
Sigma Models (GLSMs) possessing (0,2) worldsheet supersymmetry are employed as
interpolations between them. This question is addressed for resolutions of the
non-compact C3/Z2xZ2 and the compact T6/Z2xZ2 orbifolds to keep track of local
and global aspects. The GLSMs associated with the non-compact orbifold with or
without torsion are to a large extent equivalent: only when expressed in the
same superfield basis, a field redefinition anomaly arises among them, which in
the orbifold limit reproduces the discrete torsion phases. Previously unknown,
novel resolution GLSMs for T6/Z2xZ2 are constructed. The GLSM associated with
the torsional compact orbifold suffers from mixed gauge anomalies, which need
to be cancelled by appropriate logarithmic superfield dependent FI-terms on the
worldsheet, signalling H-flux due to NS5-branes supported at the exceptional
cycles.
