We present SAURON integral-field observations of a sample of 12 mid-to-high-inclination disc galaxies, to unveil hidden bars on the basis of their kinematics, i.e. the correlation between velocity and h3 profiles, and to establish their degree of cylindrical rotation. For the latter, we introduce a method to quantify cylindrical rotation that is robust against inner disc components. We confirm high levels of cylindrical rotation in boxy/peanut bulges, but also observe this feature in a few galaxies with rounder bulges. We suggest that these are also barred galaxies with end-on orientations. Re-analysing published data for our own Galaxy using this new method, we determine that the Milky Way bulge is cylindrically rotating at the same level as the strongest barred galaxy in our sample. Finally, we use self-consistent three-dimensional N-body simulations of bar-unstable discs to study the dependence of cylindrical rotation on the bar's orientation and host galaxy inclination.