The nature of the linear dichroism in the near-edge X-ray absorption fine structure (NEXAFS) spectra of linear n-alkanes is a matter of long-standing controversy. Linear dichroism in the carbon 1s -> sigma*(C-C) transition has been interpreted within a building block model and a molecular orbital model, leading to two different descriptions for the angular dependence of this feature. When used for measurement of molecular orientation, the application of these two different models will lead to different results. We have explored the linear dichroism in the carbon is NEXAFS spectra of single crystals of the linear n-alkane hexacontane (n-C60H122). An analysis of the angular dependence in this spectrum shows that the transition dipole moment associated with the carbon 1s -> sigma*(C-C) transition is oriented along the macromolecular chain axis, contradicting the predictions of the building block model. However, other transitions are observed in the sigma*(C-H), and the sigma*(C-C) bands that are orthogonal to the dominant transitions for each band. We also observe that radiation damage can be manifest in the form of molecular reorientation in highly ordered organic thin films.