We present an analysis of nine S0-Sb galaxies which have (photometric) bulges consisting of two distinct components. The outer component is a flattened, kinematically cool, disc-like structure: a `discy pseudo-bulge'. Embedded inside is a rounder, kinematically hot spheroidal structure: a `classical bulge'. This indicates that pseudo-bulges and classical bulges are not mutually exclusive phenomena: some galaxies have both. The discy pseudo-bulges almost always consist of an exponential disc (scalelengths = 125-870 pc, mean size ˜440 pc) with one or more disc-related subcomponents: nuclear rings, nuclear bars, and/or spiral arms. They constitute 11-59 per cent of the galaxy stellar mass (mean PB/T = 0.33), with stellar masses ˜7 × 109-9 × 1010 M⊙. The classical-bulge components have Sérsic indices of 0.9-2.2, effective radii of 25-430 pc and stellar masses of 5 × 108-3 × 1010 M⊙; they are usually <10 per cent of the galaxy's stellar mass (mean B/T = 0.06). The classical bulges do show rotation, but are clearly kinematically hotter than the discy pseudo-bulges. Dynamical modelling of three systems indicates that velocity dispersions are isotropic in the classical bulges and equatorially biased in the discy pseudo-bulges. In the mass-radius and mass-stellar mass density planes, classical-bulge components follow sequences defined by ellipticals and (larger) classical bulges. Discy pseudo-bulges also fall on this sequence; they are more compact than large-scale discs of similar mass. Although some classical bulges are quite compact, they are as a class clearly distinct from nuclear star clusters in both size and mass; in at least two galaxies they coexist with nuclear clusters. Since almost all the galaxies in this study are barred, they probably also host boxy/peanut-shaped bulges (vertically thickened inner parts of bars). NGC 3368 shows isophotal evidence for such a zone just outside its discy pseudo-bulge, making it a clear case of a galaxy with all three types of `bulge'.