Thin films of a few layers of graphene obtained by solid-state graphitization from 6H-SiC(0001) substrates have been studied by X-ray photoelectron spectroscopy (XPS) and X-ray photoelectron diffraction (XPD). The C1s core-level was resolved into components, which were associated with carbon from bulk SiC, carbon from graphene and carbon at the graphene/6H-SiC(0001)interface. Then, the intensity of each of these components was recorded as a function of the polar (azimuth) angle for several azimuth (polar) angles. These XPD measurements provide crystallographic information which clearly indicates that the graphene sheets are organized in a graphite-like structure on 6H-SiC(0001), an organisation that results from the shrinking of the 6H-SiC(0001) lattice after Si depletion. Finally the decoupling of graphene from the 6H-SiC(0001) substrate by oxygen intercalation was studied from the XPS point of view.