In Paper I of this series we introduced a volume-limited parent sample of 871 galaxies from which we extracted the ATLAS 3D sample of 260 early-type galaxies (ETGs). In Papers II and III we classified the ETGs using their stellar kinematics, in a way that is nearly insensitive to the projection effects, and we separated them into fast and slow rotators. Here we look at galaxy morphology and note that the edge-on fast rotators generally are lenticular galaxies. They appear like spiral galaxies with the gas and dust removed, and in some cases are flat ellipticals (E5 or flatter) with discy isophotes. Fast rotators are often barred and span the same full range of bulge fractions as spiral galaxies. The slow rotators are rounder (E4 or rounder, except for counter-rotating discs) and are generally consistent with being genuine, namely spheroidal-like, elliptical galaxies. We propose a revision to the tuning-fork diagram by Hubble as it gives a misleading description of ETGs by ignoring the large variation in the bulge sizes of fast rotators. Motivated by the fact that only one third (34 per cent) of the ellipticals in our sample are slow rotators, we study for the first time the kinematic morphology-density T-Σ relation using fast and slow rotators to replace lenticulars and ellipticals. We find that our relation is cleaner than using classic morphology. Slow rotators are nearly absent at the lowest density environments [per cent] and generally constitute a small fraction [f(SR) ≈ 4 per cent] of the total galaxy population in the relatively low-density environments explored by our survey, with the exception of the densest core of the Virgo cluster [f(SR) ≈ 20 per cent]. This contrasts with the classic studies that invariably find significant fractions of (misclassified) ellipticals down to the lowest environmental densities. We find a clean log-linear relation between the fraction f(Sp) of spiral galaxies and the local galaxy surface density Σ 3, within a cylinder enclosing the three nearest galaxies. This holds for nearly four orders of magnitude in the surface density down to Σ 3≈ 0.01Mpc -2, with f(Sp) decreasing by 10 per cent per dex in Σ 3, while f(FR) correspondingly increases. The existence of a smooth kinematic T-Σ relation in the field excludes processes related to the cluster environment, like e.g. ram-pressure stripping, as main contributors to the apparent conversion of spirals into fast rotators in low-density environments. It shows that the segregation is driven by local effects at the small-group scale. This is supported by the relation becoming shallower when using a surface density estimator Σ 10 with a cluster scale. Only at the largest densities in the Virgo core does the f(Sp) relation break down and steepen sharply, while the fraction of slow rotators starts to significantly increase. This suggests that a different mechanism is at work there, possibly related to the stripping of the gas from spirals by the hot intergalactic medium in the cluster core and the corresponding lack of cold accretion.