Native cellulose in higher plants forms crystalline fibrils a few nm across, with a substantial fraction of their glucan chains at the surface. The accepted crystal structures feature a flat-ribbon 21 helical chain conformation with every glucose residue locked to the next by hydrogen bonds from O-3' to O-5 and from O-2 to O-6'. Using solid-state NMR spectroscopy we show that the surface chains have a different C-6 conformation so that O-6 is not in the correct position for the hydrogen bond from O-2. We also present evidence consistent with a model in which alternate glucosyl residues are transiently or permanently twisted away from the flat-ribbon conformation of the chain, weakening the O-3' - 0-5 hydrogen bond. Previous molecular modelling and the modelling studies reported here indicate that this 'translational' chain conformation is energetically feasible and does not preclude binding of the surface chains to the interior chains, because the surface chains share the axial repeat distance of the 21 helix. Reduced intramolecular hydrogen bonding allows the surface chains to form more hydrogen bonds to external molecules in textiles, wood, paper and the living plant.