  We present a comprehensive reassessment of the state of Interacting Dark
Energy (IDE) cosmology, namely models featuring a non-gravitational interaction
between Dark Matter (DM) and Dark Energy (DE). To achieve high generality, we
extend the dark sector physics by considering two different scenarios: a
non-dynamical DE equation of state $w_0\neq-1$, and a dynamical
$w(a)=w_0+w_a(1-a)$. In both cases, we distinguish two different physical
regimes resulting from a phantom or quintessence equation of state. To
circumvent early-time superhorizon instabilities, the energy-momentum transfer
should occur in opposing directions within the two regimes, resulting in
distinct phenomenological outcomes. We study quintessence and phantom
non-dynamical and dynamical models in light of two independent Cosmic Microwave
Background (CMB) experiments - the Planck satellite and the Atacama Cosmology
Telescope. We analyze CMB data both independently and in combination with
Supernovae (SN) distance moduli measurements from the Pantheon-Plus catalog and
Baryon Acoustic Oscillations (BAO) from the SDSS-IV eBOSS survey. Our results
update and extend the state-of-the-art analyses, significantly narrowing the
parameter space allowed for these models and limiting their overall ability to
reconcile cosmological tensions. Although considering different combinations of
data leaves some freedom to increase $H_0$ towards the value measured by the
SH0ES collaboration, our most constraining dataset (CMB+BAO+SN) indicates that
fully reconciling the tension solely within the framework of IDE remains
challenging.
