  Expanding upon earlier results [arXiv:1702.02861], we present a compendium of
$\sigma$-models associated with integrable deformations of AdS$_5$ generated by
solutions to homogenous classical Yang-Baxter equation. Each example we study
from four viewpoints: conformal (Drinfeld) twists, closed string gravity
backgrounds, open string parameters and proposed dual noncommutative (NC) gauge
theory. Irrespective of whether the deformed background is a solution to
supergravity or generalized supergravity, we show that the open string metric
associated with each gravity background is undeformed AdS$_5$ with constant
open string coupling and the NC structure $\Theta$ is directly related to the
conformal twist. One novel feature is that $\Theta$ exhibits "holographic
noncommutativity": while it may exhibit non-trivial dependence on the
holographic direction, its value everywhere in the bulk is uniquely determined
by its value at the boundary, thus facilitating introduction of a dual NC gauge
theory. We show that the divergence of the NC structure $\Theta$ is directly
related to the unimodularity of the twist. We discuss the implementation of an
outer automorphism of the conformal algebra as a coordinate transformation in
the AdS bulk and discuss its implications for Yang-Baxter $\sigma$-models and
self-T-duality based on fermionic T-duality. Finally, we comment on
implications of our results for the integrability of associated open strings
and planar integrability of dual NC gauge theories.
