  We propose a new probe of inflationary gravitational waves (IGWs): the
cross-correlation of the lensing of inflationary $B$-mode polarization with a
large-scale structure (LSS) tracer, which can also be a cosmic microwave
background (CMB) lensing map. This is equivalent to measuring a three-point
function of two CMB $B$-modes and an LSS tracer. We forecast expected
$1\,\sigma$ constraints on the tensor-to-scalar ratio $r$, albeit with a
simplistic foreground treatment, and find constraints of $\sigma_r \simeq 7
\times 10^{-3}$ from the correlation of CMB-S4-Deep $B$-mode lensing and LSST
galaxies, $\sigma_r \simeq 5 \times 10^{-3}$ from the correlation of
CMB-S4-Deep $B$-mode lensing and CMB-S4-Deep CMB lensing, and $\sigma_r \simeq
10^{-2}$ from the correlation of LiteBIRD $B$-mode lensing and CMB-S4-Wide
lensing. Because this probe is inherently non-Gaussian, simple Gaussian
foregrounds will not produce any biases to the measurement of $r$. While a
detailed investigation of non-Gaussian foreground contamination for different
cross-correlations will be essential, this observable has the potential to be a
useful probe of IGWs, which, due to different sensitivity to many potential
sources of systematic errors, can be complementary to standard methods for
constraining $r$.
