  We look at the morphology of fast and slow rotator early-type galaxies.
Edge-on fast rotators are lenticular galaxies. They appear like spiral galaxies
with the gas and dust removed, and in some cases are flat ellipticals with
disky isophotes. Fast rotators are often barred and span the same full range of
bulge fractions as spiral galaxies. The slow rotators are rounder 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. We study for the first time the
kinematic morphology-density T-Sigma 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 [f(SR)<2%] and generally constitute a small fraction [f(SR)~4%] of
the total galaxy population in the environments explored by our survey, with
the exception of the densest core of the Virgo cluster [f(SR)~20%]. We find a
clean log-linear relation between the fraction f(Sp) of spiral galaxies and the
local galaxy surface density. The existence of a smooth kinematic T-Sigma
relation in the field excludes processes related to the cluster environment 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. Only at the largest densities in the Virgo
core does the f(Sp) relation break down and steepens sharply, while the
fraction of slow-rotators starts to significantly increase. This suggests that
a different mechanism is at work there. (Abridged)
