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We present detailed morphological, photometric, and stellar-kinematicanalyses of the central regions of two massive, early-type barred galaxies withnearly identical large-scale morphologies. Both have large, strong bars withprominent inner photometric excesses that we associate with boxy/peanut-shaped(B/P) bulges; the latter constitute ~ 30% of the galaxy light. Inside its B/Pbulge, NGC 4608 has a compact, almost circular structure (half-light radius R_eapprox. 310 pc, S\'ersic n = 2.2) we identify as a classical bulge, amountingto 12.1% of the total light, along with a nuclear star cluster (R_e ~ 4 pc).NGC 4643, in contrast, has a nuclear disc with an unusual broken-exponentialsurface-brightness profile (13.2% of the light), and a very small spheroidalcomponent (R_e approx. 35 pc, n = 1.6; 0.5% of the light). IFU stellarkinematics support this picture, with NGC 4608's classical bulge slowlyrotating and dominated by high velocity dispersion, while NGC 4643's nucleardisc shows a drop to lower dispersion, rapid rotation, V-h3 anticorrelation,and elevated h4. Both galaxies show at least some evidence for V-h3 correlationin the bar (outside the respective classical bulge and nuclear disc), inagreement with model predictions. Standard 2-component (bulge/disc)decompositions yield B/T ~ 0.5-0.7 (and bulge n > 2) for both galaxies. Thisoverestimates the true "spheroid" components by factors of four (NGC 4608) andover 100 (NGC 4643), illustrating the perils of naive bulge-disc decompositionsapplied to massive barred galaxies.

Composite Bulges -- II. Classical Bulges and Nuclear Discs in Barred Galaxies: The Contrasting Cases of NGC 4608 and NGC 4643