The Fe 2p X-ray absorption spectra of single crystal gillespite, BaFeSi₄O₁₀, show a strong linear dichroism, i.e. a large difference in the absorption when measured with the polarization of the X-rays either parallel or perpendicular to the plane of the FeO₄ group. The isotropic spectrum, obtained from measurement at the ‘magic angle’, and the polarization dependent spectra have been compared to atomic multiplet calculations and show an excellent agreement with theory. Analysis of the branching ratio, the linear dichroism, and the detailed peak structure confirms that the ⁵A₁ level is the ground state at room temperature and pressure. The ⁵B₂ level is sufficiently low in energy that a distortion of the electronic charge density, induced by increased pressure, may result in a ⁵B₂ ground state.