The problem of the experimental and theoretical determination of magnetic anisotropy in isolated molecular spin clusters is addressed here. To this end, the case of molecular Cr7Ni rings sublimated in ultrahigh vacuum conditions and assembled in an ordered fashion on Au(111) surface is addressed and investigated using X-ray magnetic dichroism (XMCD) and theoretical calculations. Fixing the experimental conditions at a temperature T = 8 K and a magnetic field of 5 T, the angular-dependence of the dichroic signal reveals an easy-axis anisotropy for the Ni magnetization along the direction perpendicular to the ring while the magnetization of the whole Cr7Ni molecule is preferentially aligned within the ring plane. These features are well reproduced by spin Hamiltonian simulations, which reflect the character of the S = 3/2 first excited multiplet, dominating at T = 8 K and 5 T. Density functional theory (DFT) calculations show that local spin orbit interactions determine an easy axis anisotropy at the Ni site while the Cr magnetic moment turns out to be more isotropic. This is the first direct observation of the interplay between the single ion and the overall magnetic anisotropy in complex (polynuclear) molecular systems.