We present spectropolarimetric observations of seven broad-line radio galaxies. We find significantly polarized broad Hα emission in four objects including two, Arp 102B and 3C 390.3, which have double-peaked line profiles. In these objects the prominent redshifted and blueshifted peaks of the broad Hα line have no clear counterparts in polarized flux. This conflicts with theoretical predictions for a relativistic line-emitting disc with an electron scattering atmosphere, one of the leading models advanced to account for the double-peaked lines. The shapes and widths of the polarized line profiles can be explained if, as expected in unified schemes, the scattering occurs near the poles of an obscuring torus. However, the observed polarization position angles favour geometries in which the scattering plane is perpendicular to the radio jet. A configuration in which Hα photons emitted by a biconical flow are scattered off the inner wall of the torus has this property, and would also produce a single-peaked scattered line profile. With the exception of 3C 227, the sample as a whole conforms to the general trend in powerful radio galaxies for the optical polarization to be aligned with the radio source axis, favouring toroidal rather than polar scattering.