  The circumgalactic medium (CGM) encodes signatures of the galaxy-formation
process, including the interaction of galactic outflows driven by stellar and
supermassive black hole (SMBH) feedback with the gaseous halo. Moving beyond
spherically symmetric radial profiles, we study the \textit{angular} dependence
of CGM properties around $z=0$ massive galaxies in the IllustrisTNG
simulations. We characterize the angular signal of density, temperature, and
metallicity of the CGM as a function of galaxy stellar mass, halo mass,
distance, and SMBH mass, via stacking. TNG predicts that the CGM is anisotropic
in its thermodynamical properties and chemical content over a large mass range,
$M_*\sim10^{10-11.5}M_\odot$. Along the minor axis directions, gas density is
diluted, whereas temperature and metallicity are enhanced. These
feedback-induced anisotropies in the CGM have a magnitude of $0.1-0.3$ dex,
extend out to the halo virial radius, and peak at Milky Way-like masses,
$M_*\sim10^{10.8}M_\odot$. In TNG, this mass scale corresponds to the onset of
efficient SMBH feedback and the production of strong outflows. By comparing the
anisotropic signals predicted by TNG versus other simulations -- Illustris and
EAGLE -- we find that each simulation produces distinct signatures and mass
dependencies, implying that this phenomenon is sensitive to the underlying
physical models. Finally, we explore X-ray emission as an observable of this
CGM anistropy, finding that future X-ray observations, including the eROSITA
all-sky survey, will be able to detect and characterize this signal,
particularly in terms of an angular modulation of the X-ray hardness.
