  Many inflationary theories predict a non-Gaussian spectrum of primordial
tensor perturbations, sourced from non-standard vacuum fluctuations, modified
general relativity or new particles such as gauge fields. Several such models
also predict a chiral spectrum in which one polarization state dominates. In
this work, we place constraints on the non-Gaussianity and parity properties of
primordial gravitational waves utilizing the Planck PR4 temperature and
polarization dataset. Using recently developed quasi-optimal bispectrum
estimators, we compute binned parity-even and parity-odd bispectra for all
combinations of CMB T-, E- and B-modes with $2\leq \ell<500$, and perform both
blind tests, sensitive to arbitrary three-point functions, and targeted
analyses of a well-motivated equilateral gravitational wave template (sourced
by gauge fields), with amplitude $f_{\rm NL}^{ttt}$. This is the first time
B-modes have been included in primordial non-Gaussianity analyses; they are
found to strengthen constraints on the parity-even sector by $\simeq 30\%$ and
dominate the parity-odd bounds, without inducing bias. We report no detection
of non-Gaussianity (of either parity), with the template amplitude constrained
to $f_{\rm NL}^{ttt}=900\pm 700$ (stable with respect to a number of analysis
variations), compared to $1300\pm1200$ in Planck 2018. The methods applied
herein can be reapplied to upcoming CMB datasets such as LiteBIRD, with the
inclusion of B-modes poised to dramatically improve future bounds on tensor
non-Gaussianity.
