Thermally induced growth of graphene on the two polar surfaces of 6H-SiC is investigated with emphasis on the initial stages of growth and interface structure. The experimental methods employed are angle-resolved valence band photoelectron spectroscopy, soft x-ray induced core-level spectroscopy, and low-energy electron diffraction. On the Si-terminated ͑0001͒ surface, the ͑6 ͱ 3 ϫ 6 ͱ 3͒R30°reconstruction is the precursor of the growth of graphene and it persists at the interface upon the growth of few layer graphene ͑FLG͒. The ͑6 ͱ 3 ϫ 6 ͱ 3͒R30°structure is a carbon layer with graphene-like atomic arrangement covalently bonded to the substrate where it is responsible for the azimuthal ordering of FLG on SiC͑0001͒. In contrast, the interaction between graphene and the C-terminated ͑0001͒ surface is much weaker, which accounts for the low degree of order of FLG on this surface. A model for the growth of FLG on SiC͕0001͖ is developed, wherein each new graphene layer is formed at the bottom of the existing stack rather than on its top. This model yields, in conjunction with the differences in the interfacial bonding strength, a natural explanation for the different degrees of azimuthal order observed for FLG on the two surfaces.