  Gravity-mode asteroseismology has significantly improved our understanding of
mixing in intermediate mass stars. However, theoretical pulsation periods of
stellar models remain in tension with observations, and it is often unclear how
the models of these stars should be further improved. Inversions provide a path
forward by directly probing the internal structure of these stars from their
pulsation periods, quantifying which parts of the model are in need of
improvement. This method has been used for solar-like pulsators, but has not
yet been applied to main-sequence gravity-mode pulsators. Our aim is to
determine whether structure inversions for gravity-mode pulsators are feasible.
We focus on the case of slowly rotating SPB stars. We computed and analyzed
dipole mode kernels for three variables pairs: $(\rho,c), (N^2,c)$, and
$(N^2,\rho)$. We assessed the potential of these kernels by predicting the
oscillation frequencies of a model after perturbing its structure. We then
tested two inversion methods, RLS and SOLA, using a model grid computed with
MESA and GYRE. We find that changing the stellar structure affects the
oscillation frequencies in a nonlinear way. The oscillation modes for which
this nonlinear dependency is the strongest are in resonance with the near-core
peak in the buoyancy frequency. The near core region of the star can be probed
with SOLA, while RLS requires fine tuning to obtain accurate results. Both RLS
and SOLA are strongly affected by the nonlinear dependencies on the structure
differences, as these methods are based on a first-order approximation. These
inversion methods need to be modified for meaningful applications of inversions
to SPB stars. Our results show that inversions of gravity-mode pulsators are
possible in principle, but that the typical inversion methods developed for
solar-like oscillators are not applicable. [abridged]
