  We study an atomic comagnetometer design based on the spin precessions of
$^{129}$Xe and $^{131}$Xe atoms in glass cells. The quadrupole splittings in
the precession spectrum of $^{131}$Xe are fully resolved, allowing a precise
determination of the magnetic-dipole precession frequency. The transverse
asymmetry of quadrupole interactions, due to both the geometry and surface
properties of the cell, characterized by a non-zero asymmetry parameter $\eta$,
modifies the dependence of the quadrupole splittings on the relative
orientation between the cell axes and the bias magnetic field, and lead to
additional corrections in the precession frequencies of $^{131}$Xe atoms. We
examine these effects both theoretically and experimentally, and develop
methods to quantify and control such shifts.
