  The environment-dependent bimodality of the distribution of stellar mass
($M_\ast$) and specific star formation rate (sSFR) of galaxies, and its
explanation in terms of the central-satellite dichotomy, form a cornerstone of
our current understanding of galaxy evolution in the hierarchical structure
formation paradigm. We revisit this framework in the IllustrisTNG simulation in
the context of the most extreme local tidal anisotropy $\alpha_{\rm peak}$
experienced by each galaxy over cosmic time, which is an excellent proxy for
environmental influence. We show that, while sharing a common monotonic
$M_\ast$-$v_{\rm peak}$ relation, central, satellite and `splashback' galaxies
define a hierarchy of increasing $\alpha_{\rm peak}$. We also find that the
sSFR of objects in small haloes unaffected by feedback from an active nucleus
typically decreases with increasing $\alpha_{\rm peak}$. Our results support an
alternate viewpoint in which a galaxy can be identified by the value of
$\alpha_{\rm peak}$; i.e., rather than being placed on the central-satellite
dichotomy, a galaxy is better classified by its location in a continuum of
tidal environments. This conceptual shift can potentially yield a more robust
understanding of galaxy evolution and the galaxy-dark matter connection, e.g.,
in accurately modelling subtle effects such as sSFR-induced secondary
clustering.
