We performed a combined theoretical and experimental investigation of the orbital magnetism and magnetocrystalline anisotropy of isolated Co and Fe adatoms on Pd(111) and Rh(111). Theoretical calculations of the spin and orbital moments are based on ab initio spin-polarized density-functional theory (DFT) including a self-consistent treatment of spin-orbit coupling. The calculations use a slab model to represent the adsorbate/substrate complex and allow for a complete structural relaxation leading to a strong inward displacement of the adatom and modest vertical and lateral relaxations in the substrate atoms. Compared to an idealized geometry where the atoms are kept on bulk lattice positions up to the surface, relaxation leads to a much stronger adatom/ligand hybridization. This is also reflected in the results for orbital moments and magnetocrystalline anisotropy energy (MAE). The enhanced hybridization leads to strong quenching of the adatom orbital moments but also to the formation of large induced spin and orbital moments in the substrate. As a consequence, we find that the substrate contribution to the MAE is much more important than estimated before on the basis of studies using an idealized geometry. We also find the surprising result that the MAE strongly depends on the adsorption site. The magnitude and even the sign of the MAE change for adatoms on face-centered cubic with respect to the ones on hexagonal close-packed hollow sites on the (111) surface. The dependence of the MAE on the combination of adatom and substrate has been analyzed in terms of the electronic structure, leading to a sound physical picture of the origin of the MAE. A fundamental problem, however, is the correct prediction of the size of the orbital moments of the adatoms. We suggest that this problem can be solved only via post-DFT corrections introducing an orbital dependence of the exchange potential. The theoretical results are compared to site-averaged, element-specific x-ray magnetic circular dichroism (XMCD) measurements. Low-temperature XMCD spectra and magnetization curves reveal weak out-of-plane anisotropy for Fe adatoms on both substrates. Interestingly, Co adatoms on Rh(111) present in-plane anisotropy with MAE of about -0.6 meV, contrary to the known out-of-plane anisotropy of Co on Pd(111) and Pt(111). The orbital to spin magnetic-moment ratio measured by XMCD shows that the Co adatoms present much stronger orbital magnetization components compared to Fe. The connection between orbital moments and MAE is discussed at the theoretical level including the contribution of the induced substrate magnetization.