  Two of the most rapidly growing observables in cosmology and astrophysics are
gravitational waves (GW) and the neutral hydrogen (HI) distribution. In this
work, we investigate the cross-correlation between resolved gravitational wave
detections and HI signal from intensity mapping (IM) experiments. By using a
tomographic approach with angular power spectra, including all projection
effects, we explore possible applications of the combination of the Einstein
Telescope and the SKAO intensity mapping surveys. We focus on three main
topics: \textit{(i)} statistical inference of the observed redshift
distribution of GWs; \textit{(ii)} constraints on dynamical dark energy models
as an example of cosmological studies; \textit{(iii)} determination of the
nature of the progenitors of merging binary black holes, distinguishing between
primordial and astrophysical origin. Our results show that: \textit{(i)} the GW
redshift distribution can be calibrated with good accuracy at low redshifts,
without any assumptions on cosmology or astrophysics, potentially providing a
way to probe astrophysical and cosmological models; \textit{(ii)} the
constrains on the dynamical dark energy parameters are competitive with IM-only
experiments, in a complementary way and potentially with less systematics;
\textit{(iii)} it will be possible to detect a relatively small abundance of
primordial black holes within the gravitational waves from resolved mergers.
Our results extend towards $\mathrm{GW \times IM}$ the promising field of
multi-tracing cosmology and astrophysics, which has the major advantage of
allowing scientific investigations in ways that would not be possible by
looking at single observables separately.
