Modulation of the polarization state of incident light in reflection microspectroscopy is used to study locally the orientations of exciton transition dipole moments of fiberlike nanostructures of J-aggregates of pseudoisocyanine dyes. The lowest exciton state assumes a wide range of directions with respect to the long axis of the fibers, from parallel to perpendicular. This behavior is correlated with the onset of polariton-like character of the local reflectance spectra. The results imply that a new structural model has to be conceived to explain the observed phenomena.