  We compute the new supersymmetric index of a large class of N=2 heterotic
compactifications with torsion, corresponding to principal two-torus bundles
over warped K3 surfaces with H-flux. Starting from a UV description as a (0,2)
gauged linear sigma-model with torsion, we use supersymmetric localization
techniques to provide an explicit expression of the index as a sum over the
Jeffrey-Kirwan residues of the one-loop determinant. We finally propose a
geometrical formula that gives the new supersymmetric index in terms of bundle
data, regardless of any particular choice of underlying two-dimensional theory.
