We have measured the full angular dependence, as a function of the direction of magnetic field, for the Zeeman splitting of individual energy states in copper nanoparticles. The g factors for spin splitting are highly anisotropic, with angular variations as large as a factor of 5. The angular dependence fits well to ellipsoids. Both the principal-axis directions and g-factor magnitudes vary between different energy levels within one nanoparticle. The variations agree quantitatively with random-matrix theory predictions which incorporate spin-orbit coupling.