  It is well known that non-trivial squeezed tensor bispectra can lead to
anisotropies in the inflationary stochastic gravitational wave (GW) background,
providing us with an alternative and complementary window to primordial
non-Gaussianities (NGs) with respect to the CMB. Previous works have
highlighted the detection prospects of parity-even tensor NGs via the GW
$I$-mode anisotropies. In this work we extend this by analysing for the first
time the additional information carried by GW $V$-mode anisotropies due to
squeezed NGs. We show that GW $V$ modes allow us to probe parity-odd squeezed
$\langle \rm tts \rangle$ and $\langle \rm ttt \rangle$ bispectra. These
bispectra break parity at the non-linear level and can be introduced by
allowing alternative symmetry breaking patterns during inflation, like those
comprised in solid inflation. Considering a BBO-like experiment, we find that a
non-zero detection of squeezed $\langle \rm tts \rangle$ parity-odd bispectra
in the $V$ modes dipole is possible without requiring any short-scale
enhancement of the GW power spectrum amplitude over the constraints set by the
CMB. We also briefly discuss the role of $V$-CMB cross-correlations. Our work
can be extended in several directions and motivates a systematic search for
polarized GW anisotropies in the next generations of GW experiments.
