We report on NMR spectroscopy, CAS based method calculations and X-ray diffraction on An$^V$ and An$^{VI}$ complexes with a neutral and slightly flexible TEDGA ligand. After checking that pNMR shifts mainly arise from pseudo-contact interactions, pNMR shifts have been analyzed considering axial and rhombic anisotropy of the actinyl magnetic susceptibilities. Results are compared to a previous study performed on [An $^{VI}$O$_2$]$^{2+}$ complexes with dipicolinic acid (DPA). It is shown that 5f$^2$ cations (Pu$^{VI}$ and Np$^V$) make very good candidates to obtain the structure of actinyl complexes in solution by $^1$H NMR spectroscopy as shown by the invariance of the magnetic properties to the equatorial ligands, conversely to the Np$^{VI}$ complexes with a 5f$^1$ configuration.