  Orbifolding two-dimensional quantum field theories by a symmetry group can
involve a choice of discrete torsion. We apply the general formalism of
`orbifolding defects' to study and elucidate discrete torsion for topological
field theories. In the case of Landau-Ginzburg models only the bulk sector had
been studied previously, and we re-derive all known results. We also introduce
the notion of `projective matrix factorisations', show how they naturally
describe boundary and defect sectors, and we further illustrate the efficiency
of the defect-based approach by explicitly computing RR charges. Roughly half
of our results are not restricted to Landau-Ginzburg models but hold more
generally, for any topological field theory. In particular we prove that for a
pivotal bicategory, any two objects of its orbifold completion that have the
same base are orbifold equivalent. Equivalently, from any orbifold theory
(including those based on nonabelian groups) the original unorbifolded theory
can be obtained by orbifolding via the `quantum symmetry defect'.
