The extent to which structured light might conceivably resolve the handedness of chiral matter is a topic of resurgent interest, and it is one that has been a challenge since the earliest days of optical vortex studies. It has not even been certain whether or not the orbital angular momentum of light could interact in such a way – though it has been established that electric quadrupole interactions enable twisted light to engage with local electronic transitions. Crucially, certain recent experiments have provided tantalizing evidence to support the existence of a chiral effect that is sensitive to handedness, against initial expectations. By detailed electrodynamic calculation, a new study has now fully identified the mechanism, and also provided an in-depth analysis of the role of electric quadrupole transition moments as they engage with the phase gradient of beams with a twisted wavefront. Focusing on single photon absorption, it emerges that the orbital angular momentum associated with the vorticity of a structured beam can indeed be exhibited in chiral effects, provided the material itself is not only chiral, but also has some structural order – which essentially limits the effect to chiral solids, poled liquid crystals, and oriented arrays of chiral nanoparticles. Circular polarization is still required, and the extent of circular dichroism proves to vary around the beam, being locally determined by the absorber orientation with reference to the beam axis. In agreement with earlier studies, and consistent with symmetry principles, the new analysis verifies that any dependence on wavefront vorticity vanishes in a freely mobile fluid. The reformulation of theory now paves the way for an extension to other kinds of chiroptical phenomena in orientationally ordered systems.