  Various types of inertial modes have been observed and identified on the Sun,
including the equatorial Rossby modes, critical-latitude modes, and
high-latitude modes. Recent observations further report a detection of
equatorially-antisymmetric radial vorticity modes which propagate in a
retrograde direction about three times faster than those of the equatorial
Rossby modes when seen in the corotating frame with the Sun. Here, we study the
properties of these equatorially-antisymmetric vorticity modes using a
realistic linear model of the Sun's convection zone. We find that they are
essentially non-toroidal, involving a substantial radial flow at the equator.
Thus, the background density stratification plays a critical role in
determining their dispersion relation. The solar differential rotation is also
found to have a significant impact by introducing the viscous critical layers
and confining the modes near the base of the convection zone. Furthermore, we
find that their propagation frequencies are strikingly sensitive to the
background superadiabaticity $\delta$ because the buoyancy force acts as an
additional restoring force for these non-toroidal modes. The observed
frequencies are compatible with the linear model only when the bulk of the
convection zone is weakly subadiabatic ($-5\times 10^{-7} \lesssim \delta
\lesssim -2.5\times 10^{-7}$). Our result is consistent with but tighter than
the constraint independently derived in a previous study ($\delta<2 \times
10^{-7}$) employing the high-latitude inertial mode. It is implied that, below
the strongly superadiabatic near-surface layer, the bulk of the Sun's
convection zone might be much closer to adiabatic than typically assumed and
may even be weakly subadiabatic.
