High spectral resolution linear polarization line profiles of the planetary nebula NGC 7027 have been obtained in the [O III], Halpha and [N II] emission lines. At positions offset 20 arcsec north and south of the bright optical knot, the [O III] line profile is up to 40 per cent linearly polarized, with a systematic increase in polarization from negative to positive velocity. Such high polarization can only arise through dust scattering, confirming the presence of a neutral dusty circumstellar halo, for which evidence had been found previously. The magnitude of the polarization and the velocity profile can plausibly be modelled by amorphous carbon dust with an expansion velocity similar to that of the molecular envelope. Over the bright knot, the [O III] fine has a linear polarization of 0.2 per cent at the line peak, but this increases to 9 per cent in a positive-velocity wing, which extends to +70 km s-1. This polarized wing is too extended to be explained by single scattering from moving dust grains, and the position angle of the polarization vector rotates by approximately 70-degrees across this polarization feature. Multiple scattering is probably occurring, perhaps with velocity doubling, as light is reflected across the central ionized cavity by dust in the dense waist found in CO maps.