We investigate the anisotropy of the optical properties of thin Fe films on GaAs(001) from first-principles calculations. Both intrinsic and magnetization-induced anisotropy are covered by studying the system in the presence of spin-orbit coupling and external magnetic fields. We use the linearized augmented plane wave method, as implemented in the WIEN2K density functional theory code, to show that the C-2v symmetric anisotropy of the spin-orbit coupling fields at the Fe/GaAs(001) interface manifests itself in the corresponding anisotropy of the optical conductivity and the polar magneto-optical Kerr effect. While their magnetization-induced anisotropy is negligible, the intrinsic anisotropy of the optical properties is significant and reflects the underlying C-2v symmetry of the Fe/GaAs(001) interface. This suggests that the effects of anisotropic spin-orbit coupling fields in experimentally relevant Fe/GaAs(001) slabs can be studied by purely optical means.