  Gravitational lensing rotation of images is predicted to be negligible at
linear order in density perturbations, but can be produced by the post-Born
lens-lens coupling at second order. This rotation is somewhat enhanced for
Cosmic Microwave Background (CMB) lensing due to the large source path length,
but remains small and very challenging to detect directly by CMB lensing
reconstruction alone. We show the rotation may be detectable at high
significance as a cross-correlation signal between the curl reconstructed with
Simons Observatory (SO) or CMB-S4 data, and a template constructed from
quadratic combinations of large-scale structure (LSS) tracers. Equivalently,
the lensing rotation-tracer-tracer bispectrum can also be detected, where LSS
tracers considered include the CMB lensing convergence, galaxy density, and the
Cosmic Infrared Background (CIB), or optimal combinations thereof. We forecast
that an optimal combination of these tracers can probe post-Born rotation at
the level of $5.7\sigma$-$6.1\sigma$ with SO and $13.6\sigma$-$14.7\sigma$ for
CMB-S4, depending on whether standard quadratic estimators or maximum a
posteriori iterative methods are deployed. We also show possible improvement up
to $21.3\sigma$ using a CMB-S4 deep patch observation with polarization-only
iterative lensing reconstruction. However, these cross-correlation signals have
non-zero bias because the rotation template is quadratic in the tracers, and
exists even if the lensing is rotation free. We estimate this bias
analytically, and test it using simple null-hypothesis simulations to confirm
that the bias remains subdominant to the rotation signal of interest. Detection
and then measurement of the lensing rotation cross-spectrum is therefore a
realistic target for future observations.
