We present a linear polarization map of the starburst galaxy M82, obtained in the light of monochromatic H-alpha. Polarization levels, approximately 10-30 per cent, are observed throughout the filamentary system and the centro-symmetric pattern formed by the polarization vectors demonstrates that the emission-line filaments are part of a huge bipolar reflection nebula. The circular symmetry of the polarization pattern requires a central illuminating source (less-than-or-equal-to 4-arcsec diameter) which, due to light travel time arguments, must have dominated the H-alpha emission of the galaxy for at least 5000 yr. This source is located in a small active zone within the central starburst region very close to the strong radio source 41.9 + 58 and the centre of the IR emission longward of 10-mu-m. Our observation that the filaments show up in polarized light implies that they must be due, at least in part, to illumination effects.