Boxy and peanut-shaped bulges are seen in about half of edge-on disc galaxies. Comparisons of the photometry and major-axis gas and stellar kinematics of these bulges to simulations of bar formation and evolution indicate that they are bars viewed in projection. If the properties of boxy bulges can be entirely explained by assuming that they are bars, then this may imply that their hosts are pure disc galaxies with no classical bulge. A handful of these bulges, including that of the Milky Way, have been observed to rotate cylindrically, i.e. with a mean stellar velocity independent of height above the disc. In order to assess whether such behaviour is ubiquitous in boxy bulges, and whether a pure disc interpretation is consistent with their stellar populations, we have analysed the stellar kinematics and populations of the boxy or peanut-shaped bulges in a sample of five edge-on galaxies. We placed slits along the major axis of each galaxy and at three offset but parallel positions to build up spatial coverage. The boxy bulge of NGC 3390 rotates perfectly cylindrically within the spatial extent and uncertainties of the data. This is consistent with the metallicity and α-element enhancement of the bulge, which are the same as in the disc. This galaxy is thus a pure disc galaxy. The boxy bulge of ESO 311−G012 also rotates very close to cylindrically. The boxy bulge of NGC 1381 is neither clearly cylindrically nor non-cylindrically rotating, but it has a negative vertical metallicity gradient and is α-enhanced with respect to its disc, suggesting a composite bulge comprised of a classical bulge and bar (and possibly a discy pseudo-bulge). The rotation of the peanut-shaped bulge of NGC 5746 is difficult to classify, but the peanut-shaped bulge of IC 4767 does not rotate cylindrically. Thus, even this relatively small sample is sufficient to demonstrate that boxy bulges display a range of rotational and population properties, indicating that they do not form a homogeneous class of object.