Context. Classical bulges in spiral galaxies are known to rotate, but the origin of this observed rotational motion is not well understood. It has been shown recently that a low-mass classical bulge (ClB) in a barred galaxy can acquire rotation by absorbing a significant fraction of the angular momentum emitted by the bar. Aims: Our aim here is to investigate whether bars can also spin up more massive ClBs during the secular evolution of the bar, and to study the kinematics and dynamics of these ClBs. Methods: We use a set of self-consistent N-body simulations to study the interaction of ClBs with a bar that forms self-consistently in the disk. We use orbital spectral analysis to investigate the angular momentum gain by the classical bulge stars. Results: We show that the ClBs gain on average 2-6% of the disk's initial angular momentum within the bar region. Most of this angular momentum gain occurs via low-order resonances, particularly 5:2 resonant orbits. A density wake forms in the ClB which corotates and aligns with the bar at the end of the evolution. The spin-up process creates a characteristic linear rotation profile and mild tangential anisotropy in the ClB. The induced rotation is small in the centre, but is significant beyond ~2 bulge half mass radii, where it leads to mass-weighted V/σ ~ 0.2, and reaches a local Vmax/σin ~ 0.5 at around the scale of the bar. The resulting V/σ is tightly correlated with the ratio of the bulge size to the bar size. In all models, a box/peanut bulge forms suggesting that composite bulges may be common. Conclusions: Bar-bulge resonant interaction in barred galaxies can provide some spin-up of massive ClBs, but the process appears to be less efficient than for low-mass ClBs. Further angular momentum transfer due to nuclear bars or gas inflow would be required to explain the observed rotation if it is not primordial.