The standard tight-binding dispersion relation for graphene and carbon nanotubes has an electron-hole symmetry. This symmetry has not been observed experimentally until recently. We discuss here the effect of the overlap between π-orbitals at neighbouring sites, belonging to different sublattices, on the dispersion relation. The e-h asymmetry increases with the degree of overlap, although this effect is visible only for energy bands far from the Fermi level. We estimate the observable effects of this asymmetry on the DOS which could be observed in conductivity measurements and explain the symmetry found in the experiment. The degree of the overlap determines also the magnitude of the magnetic moment induced in the nanotube by the Aharonov-Bohm magnetic flux.