Observed anti-correlation between V/σ and h₃. Left: Local V/σ − h₃ relation for every spectrum within the effective radius and with an error on h₃ smaller than 0.05 for M3G galaxies. Red contours are for NRR galaxies, and blue contours are for RR galaxies, and they are based on logarithmic number counts starting from 0.25 with a step of 0.5. Straight lines show the slopes of V/σ − h₃ distributions estimated by the ξ₃ parameter (see Section 4.3 for details) for M3G galaxies separated into RR (blue) and NRR (red), BCGs (black solid) and non-BCGs (black dashed), and three classes based on the kinematic misalignment angle (see Table S2): ‘oblate’ (violet dashed), ‘triaxial’ (violet dotted) and ‘prolate’ (violet solid). The values for the local ETGs from the ATLAS^3D survey, also separated into RR and NRR galaxies, are plotted in green. Right: Distribution of individual ξ₃ values as a function of the normalised spin parameter where the value of 0.31 separates slow rotators from fast rotators (vertical dashed line). The M3G galaxies are divided into BCGs (red) and non-BCGs (blue) and shown with large red and blue symbols, respectively, while the small light green symbols are for local ETGs from the ATLAS^3D survey. Dark red hexagons are from Loubser et al. (2022). The error bars on M3G values are typically smaller than the symbols size.