We present detailed morphological, photometric, and stellar-kinematic analyses of the central regions of two massive, early-type barred galaxies with nearly identical large-scale morphologies. Both have large, strong bars with prominent inner photometric excesses that we associate with boxy/peanut-shaped (B/P) bulges; the latter constitute ∼30 per cent of the galaxy light. Inside its B/P bulge, NGC 4608 has a compact, almost circular structure (half-light radius Re ≍ 310 pc, Sérsic n = 2.2) we identify as a classical bulge, amounting to 12.1 per cent of the total light, along with a nuclear star cluster (Re ∼ 4 pc). NGC 4643, in contrast, has a nuclear disc with an unusual broken-exponential surface-brightness profile (13.2 per cent of the light), and a very small spheroidal component (Re ≍ 35 pc, n = 1.6; 0.5 per cent of the light). IFU stellar kinematics support this picture, with NGC 4608's classical bulge slowly rotating and dominated by high velocity dispersion, while NGC 4643's nuclear disc shows a drop to lower dispersion, rapid rotation, V-h3 anticorrelation, and elevated h4. Both galaxies show at least some evidence for V-h3correlation in the bar (outside the respective classical bulge and nuclear disc), in agreement with model predictions. Standard two-component (bulge/disc) decompositions yield B/T ∼ 0.5-0.7 (and bulge n > 2) for both galaxies. This overestimates the true 'spheroid' components by factors of 4 (NGC 4608) and over 100 (NGC 4643), illustrating the perils of naive bulge-disc decompositions applied to massive barred galaxies.