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We study nine S0-Sb galaxies with (photometric) bulges consisting of two distinct components. The outer component is a flattened, kinematically cool, disklike structure: a "disky pseudobulge". Embedded inside is a rounder, kinematically hot spheroid: a "classical bulge". This indicates that pseudobulges and classical bulges are not mutually exclusive: some galaxies have both. The disky pseudobulges almost always have an exponential disk (scale lengths = 125-870 pc, mean $\sim 440$ pc) with disk-related subcomponents: nuclear rings, bars, and/or spiral arms. They constitute 11-59% of the galaxy stellar mass (mean PB/T = 0.33), with stellar masses $\sim 7 \times 10^{9}$-$9 \times 10^{10} M_{\odot}$. Classical-bulge components have Sersic indices of 0.9-2.2, effective radii of 25-430 pc and stellar masses of $5 \times 10^{8}$-$3 \times 10^{10} M_{\odot}$ (usually < 10% of the galaxy's stellar mass; mean B/T = 0.06). The classical bulges show rotation, but are kinematically hotter than the disky pseudobulges. Dynamical modeling of three systems indicates that velocity dispersions are isotropic in the classical bulges and equatorially biased in the disky pseudobulges. In the mass--radius and mass--stellar mass density planes, classical-bulge components follow sequences defined by ellipticals and (larger) classical bulges. Disky pseudobulges also fall on this sequence; they are more compact than similar-mass large-scale disks. Although some classical bulges are quite compact, they are distinct from nuclear star clusters in both size and mass, and coexist with nuclear clusters in at least two galaxies. 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 evidence for such a zone outside its disky pseudobulge, making it a galaxy with all three types of "bulge".