Active galaxies are thought to be powered by the accretion of gas onto a central massive black hole. Seyfert galaxies - the most common examples of nearby active galaxies - are separated into two classes based on their emission line widths. Seyfert 1 galaxies exhibit broad emission lines that are attributed to ionized gas within 1 pc of the black hole, whereas the spectra of Seyfert 2 galaxies show only narrower emission lines, believed to originate from a much larger region around the core. The 'unified model' for Seyfert galaxies attributes these differences to the presence of a dusty torus of dense molecular gas surrounding the black hole: the orientation of Seyfert 2 galaxies is such that the broad-line region is obscured. The detection in the polarization spectrum of broad emission lines scattered into our line of sight by free electrons in NGC1068 (the prototypical Seyfert 2 galaxy) and other Seyfert 2 galaxies has strengthened this view, but all of these galaxies were subject to selection biases. Here we report the results of a systematic search for polarized broad emission lines in a well defined sample of Seyfert 2 galaxies. We show that the ability to detect scattered broad emission lines is related to the far-infrared colours, in the manner predicted by the unified model.