  Buoyancy-restored modes inside neutron stars depend sensitively on both the
microphysical (e.g., composition and entropy gradients) and macrophysical
(e.g., stellar mass and radius) properties of the star. Asteroseismology
efforts for $g$-modes are therefore particularly promising avenues for
recovering information concerning the nuclear equation of state. In this work
it is shown that the overall low-temperature $g$-space consists of multiple
groups corresponding to different classes of equation of state (e.g., hadronic
vs. hybrid). This is in contrast to the case of pressure-driven modes, for
example, which tend to follow a universal relation regardless of microphysical
considerations. Using a wide library of currently-viable equations of state,
perturbations of static, stratified stars are calculated in general relativity
to demonstrate in particular how $g$-space groupings can be classified
according to the mean mass density, temperature, central speed of sound, and
tidal deformability. Considering present and future observations regarding
gravitational waves, accretion outbursts, quasi-periodic oscillations, and
precursor flashes from gamma-ray bursts, it is shown how one might determine
which group the $g$-modes belong to.
