At high pressure (>3300 bar) polyethylene shows a hexagonal phase between the orthorhombic phase and the liquid phase. This first NMR investigation of polyethylene in its hexagonal phase shows that the mobility of the molecular chains is very high. Deuteron quadrupole echo spectra have been used to determine the character and rate of local reorientations of the molecular chains. The chains perform rapid axial reorientations, similar to those observed in the rotator phases occurring in the long normal alkanes. From proton rotating frame relaxation data we derive a high rate of translational diffusion in the chain direction. These observations contribute to the understanding of the important role of the hexagonal phase in the growth of chain-extended polyethylene crystals.