  Low-ionization-state Mg II gas has been extensively studied in quasar
sightline observations to understand the cool, $\sim$$10^4$ K gas in the
circumgalactic medium. Motivated by recent observations showing that the Mg II
gas around low-redshift galaxies has significant angular momentum, we use the
high-resolution EAGLE cosmological simulation to analyze the morphological and
rotation structures of the $z\approx0.3$ circumgalactic Mg II gas and examine
how they change with the host galaxy properties. Around star-forming galaxies,
we find that the Mg II gas has an axisymmetric instead of a spherical
distribution, and the axis of symmetry aligns with that of the Mg II gas
rotation. A similar rotating structure is less commonly found in the small
sample of simulated quiescent galaxies. We also examine how often Mg II gas
around galaxies selected using a line-of-sight velocity cut includes gas
physically outside of the virial radius ($r_\mathrm{vir}$). For example, we
show that at an impact parameter of 100 pkpc, a $\pm500$ km s$^{-1}$ velocity
cut around galaxies with stellar masses of $10^9$-$10^{9.5}\mathrm{M_\odot}$
($10^{10}$-$10^{10.5}\mathrm{M_\odot}$) selects Mg II gas beyond the virial
radius 80% (6%) of the time. Because observers typically select Mg II gas
around target galaxies using such a velocity cut, we discuss how this issue
affects the study of circumgalactic Mg II gas properties, including the
detection of corotation. While the corotating Mg II gas generally extends
beyond $0.5r_\mathrm{vir}$, the Mg II gas outside of the virial radius
contaminates the corotation signal and makes observers less likely to conclude
that gas at large impact parameters (e.g., $\gtrsim0.25 r_\mathrm{vir}$) is
corotating.
