The magnetic properties of small diameter semiconducting single-walled carbon nanotubes have been recently predicted to depend sensitively on structural details that can be ignored for larger diameters. We confirm this dependence by directly measuring the magnetic alignment of several species of carbon nanotubes in aqueous suspension using polarized resonant photoluminescence. These data show evidence for asymmetry between nanotube chirality subclasses and are used to construct a model predicting nanotube magnetic responses for arbitrary semiconducting chirality.