One quarter of all nearby early-type galaxies (ETGs) outside Virgo host a disc/ring of HI with size from a few to tens of kpc and mass up to ~1e+9 solar masses. Here we investigate whether this HI is related to the presence of a stellar disc within the host making use of the classification of ETGs in fast and slow rotators (FR/SR). We find a large diversity of HI masses and morphologies within both families. Surprisingly, SRs are detected as often, host as much HI and have a similar rate of HI discs/rings as FRs. Accretion of HI is therefore not always linked to the growth of an inner stellar disc. The weak relation between HI and stellar disc is confirmed by their frequent kinematical misalignment in FRs, including cases of polar and counterrotating gas. In SRs the HI is usually polar. This complex picture highlights a diversity of ETG formation histories which may be lost in the relative simplicity of their inner structure and emerges when studying their outer regions. We find that LCDM hydrodynamical simulations have difficulties reproducing the HI properties of ETGs. The gas discs formed in simulations are either too massive or too small depending on the star formation feedback implementation. Kinematical misalignments match the observations only qualitatively. The main point of conflict is that nearly all simulated FRs and a large fraction of all simulated SRs host corotating HI. This establishes the HI properties of ETGs as a novel challenge to simulations.
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