We report an experimental and first-principles study of the thermal decomposition of 6H-SiC wafers, yielding graphite on the Si-terminated face and carbon nanotubes on the C-terminated face. The asymmetry of the carbon structure formation mechanisms is rationalized in terms of the different termination geometries of the opposite SiC faces. First-principles modeling reveals that horizontal, xr-delocalized carbon structures form on the Si-terminated face. The bonding network geometry of the C-terminated face favors instead the formation of vertically oriented carbon structures, which can be interpreted as nanotube lateral wall precursors.