We present optical and near-IR spectropolarimetry of the type 2 infrared-luminous galaxy IRAS 23060+0505, which clearly shows both broad Hα and Paalpha in polarized flux. The full widths at half maximum (FWHM) of these lines are consistent with that of Paalpha in total flux as measured by Hines. From the ratio of the polarized and total broad Hα fluxes, we calculate the intrinsic polarization to be ~25 per cent, assuming that the scattering geometry is the same for both the continuum and broad-line radiation. We model the flux density and polarization characteristics as a combination of scattered continuum and a dichroic view of the near-IR-emitting regions through the postulated torus, which obscures the type 1 nucleus from the line of sight. The extinction along the path of the scattered radiation is A_V~2.9 mag, whilst that of the direct view to the near-IR emission region is 20 mag, and to the BLR is ~30 mag, substantially higher than previous estimates. The presence of high extinction along the scattering path may have strong implications for the observability of scattered optical broad lines in other objects. The calculated broad Hα luminosity is (2.3+/-0.3)x10^44 erg s^-1, consistent with that of QSOs.