tokens: size equal to the sum of the number of HDs and CRs. This results in 118 HDs, 47 CRs, and 165 NDs (see Table 1 ). We show the distribution of the three samples in the plane in the right-hand panel of Fig. 1 , which is a zoom-in figure with galaxies with and . As can be seen, all three sub-populations have roughly similar sSFR, with CRs being slightly lower than the other two. In addition, we have also checked and confirmed that the total halo masses of the three galaxy samples are roughly within the same range. Table 1. Number of galaxies in different samples. Columns (1): All z = 0 central galaxies from the IllustrisTNG simulation; also the parent sample of Columns (2)−(5). Column (2): late-type galaxies; Column (3): selected hot disc galaxies (HDs); Column (4): selected counter-rotating disc galaxies (CRs); and Col.(5): control-sample normal disc galaxies (NDs). See Section 2.2 for definition and selection criteria. In Fig. 2 , we show the distributions of the three orbital fractions ( f cold , f hot , and f ccd ) for general late-type galaxies and the three selected samples (HDs, CRs, and NDs). As can be seen, the three samples exhibit distinctive differences in cold, hot, and counter-rotating cold orbital fractions: NDs, which host the highest fractions in f cold , also possess the lowest fractions in f hot and f ccd , while HDs indeed have lower fractions in f cold and f ccd , and CRs have the lowest fractions in f cold and moderate fractions in f hot . This also demonstrates that the three types of galaxies are intrinsically different from each other. Figure 2. Distributions of f cold , f hot , and f ccd (from left to right, see Section 2.1 for definitions) for late-type galaxies (black), NDs (blue), HDs (red), and CRs (green). Black dashed lines represent f cold = 0.4, f hot = 0.31, and f ccd = 0.2, which are used to classify NDs, HDs, and CRs, respectively. 3 WHAT DO THEY LOOK LIKE? 3.1 Morphology of stars and gas To discuss the visually apparent differences in morphologies between HDs, CRs, and NDs, we first present in Fig. 3 the stellar and H i mass density maps from both edge-on and face-on views for typical HDs (the first 2 columns), CRs (the middle 2 columns), and NDs (the latter 2 columns). As can be seen, the two NDs have the flattest stellar and gaseous disc morphologies with their spin axes well-aligned. In comparison, however, both CRs have less flat stellar and gaseous discs, the spins of which exhibit very large misalignment (∼180°); the two HDs possess much thicker stellar discs and display the least established gaseous discs among all three types. Interestingly, very often the optical images of NDs and CRs show clear extended spiral patterns which are absent in the HD samples. Figure 3. Examples of two HDs (ID: 582800, 512569), two CRs (ID: 486917, 532146), and two NDs (ID: 462986, 606707) from left to right. For each example, we show from the top to the bottom rows: (1) the face-on view of the stellar mass density map,
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