  We explore and corroborate, by working out explicit examples, the
effectiveness of cross-correlating stochastic gravitational wave background
anisotropies with CMB temperature fluctuations as a way to establish the
primordial nature of a given gravitational wave signal. We consider the case of
gravitational wave anisotropies induced by scalar-tensor-tensor primordial
non-Gaussianity. Our analysis spans anisotropies exhibiting different angular
behaviours, including a quadrupolar dependence. We calculate the expected
uncertainty on the non-linearity parameter $F_{\rm NL}$ obtained as a result of
cross-correlation measurements for several proposed experiments such as the
ground-based Einstein Telescope, Cosmic Explorer, and the space-based Big-Bang
Observer. As a benchmark for future survey planning, we also calculate the
theoretical, cosmic-variance-limited, error on the non-linearity parameter.
