The orientational behaviour of ortho-H2 and para-D2 molecules adsorbed on graphite has been studied by NMR from 0.3 to 12 K at coverages where the molecules form a [Formula: see text] registered lattice. We have observed that for 90% ortho-H2, the splitting of the high-temperature NMR doublet increases rapidly near 0.6 K and additional structure appears. We have identified this as a transition to the orientationally ordered pinwheel phase: one of the phases predicted by the mean field theory of Harris and Berlinsky. In contrast, the spectrum of 90% para-D2 evolves smoothly to a broad, weak structure as the temperature decreases, showing no evidence of ordering down to 0.3 K. This very different behaviour is unexpected, given that in the bulk solid both isotopes order at comparable temperatures. From the splitting of the high-temperature NMR doublets, we have deduced the crystal fields Vc and the effective quadrupole coupling constants Γ. For D2, Γ is somewhat less than the rigid lattice value, as expected in the presence of translational zero-point motion. However, for H2, Γ is anomalously small. The ortho–para conversion rates of H2 and D2 on our Grafoil substrate have also been measured.