We present Hα spectropolarimetry observations of a sample of 23 Herbig Ae/Be stars. A change in the linear polarization across Hα is detected in a large fraction of the objects, which indicates that the regions around Herbig stars are flattened (disc-like) on small scales. A second outcome of our study is that the spectropolarimetric signatures for the Ae stars differ from those of the Herbig Be stars, with characteristics changing from depolarization across Hα in the Herbig Be stars, to line polarizations in the Ae group. The frequency of depolarizations detected in the Herbig Be stars (seven out of 12) is particularly interesting as, by analogy with classical Be stars, it may be the best evidence to date that the higher-mass Herbig stars are surrounded by flattened structures. For the Herbig Ae stars, nine out of 11 show a line polarization effect that can be understood in terms of a compact Hα emission that is itself polarized by a rotating disc-like circumstellar medium. The spectropolarimetric difference between the Herbig Be and Ae stars may be the first indication that there is a transition in the Hertzsprung–Russell diagram from magnetic accretion at spectral type A to disc accretion at spectral type B. Alternatively, the interior polarized line emission apparent in the Ae stars may be masked in the Herbig Be stars owing to their higher levels of Hα emission.