We study synergies between three promising methods to measure $22$ and suffer from
interloper problems even for spectroscopic surveys. Intensity mapping of the
21-cm signal can cover large volumes with exquisite fidelity, but is limited
both by loss of information to foreground cleaning and by lack of knowledge of
the mean signal. Cosmic microwave background (CMB) lensing is theoretically
very clean, but ultimately measures just the projected variations in density.
We find that cross-correlation between optical and radio can significantly
improve the measurement of growth rate. Combining these with the CMB provides a
promising avenue to detecting modified gravity at high redshifts, in particular
by independently probing the Weyl and Newtonian potentials and by strengthening
control of systematics. Assuming a Stage 4 CMB-survey, we find that
cross-correlating a Stage {\sc ii} 21-cm survey with DESI quasars could enable
measurements of the growth rate $f\sigma_8$ and the gravitational slip $\gamma$
at sub 3\% and sub 8\% levels at $z = 3, 4$, representing a factor of 4 and 8
improvement over constraints obtainable from DESI quasars alone. Similarly,
cross-correlating 21-cm data with a LBG survey to $m_{UV}<24.5$ over 1000
square degrees will make possible $f\sigma_8$ and $\gamma$ measurements at
close to 1\% at $z = 3$ and 3\% at $z = 4$, and improve similar constraints at
$z = 5$ by close to a factor of 3 to sub-10\% precision, enabling us to test
the predictions of general relativity at large scales.