We study the evidence for a diversity of formation processes in early-type galaxies by presenting the first complete volume-limited sample of slow rotators with both integral-field kinematics from the ATLAS3D Project and high spatial resolution photometry from the Hubble Space Telescope. Analysing the nuclear surface brightness profiles of 12 newly imaged slow rotators, we classify their light profiles as core-less, and place an upper limit to the core size of ~10 pc. Considering the full magnitude and volume-limited ATLAS3D sample, we correlate the presence or lack of cores with stellar kinematics, including the proxy for the stellar angular momentum and the velocity dispersion within one half-light radius, stellar mass, stellar age, $alpha$-element abundance, and age and metallicity gradients. More than half of the slow rotators have core-less light profiles, and they are all less massive than $10^{11}$ Msun. Core-less slow rotators show evidence for counter-rotating flattened structures, have steeper metallicity gradients, and a larger dispersion of gradient values than core slow rotators. Our results suggest that core and core-less slow rotators have different assembly processes, where the former are the relics of massive dissipation-less merging in the presence of central supermassive black holes. Formation processes of core-less slow rotators are consistent with accretion of counter-rotating gas or gas-rich mergers of special orbital configurations, which lower the final net angular momentum of stars, but support star formation. We also highlight core fast rotators as galaxies that share properties of core slow rotators and core-less slow rotators. Formation processes similar to those for core-less slow rotators can be invoked to explain the assembly of core fast rotators, with the distinction that these processes form or preserve cores.[Abridged]
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