  Measurement of the cosmic microwave background (CMB) bispectrum, or
three-point correlation function, has now become one of the principle efforts
in early-Universe cosmology. Here we show that there is a odd-parity component
of the CMB bispectrum that has been hitherto unexplored. We argue that
odd-parity temperature-polarization bispectra can arise, in principle, through
weak lensing of the CMB by chiral gravitational waves or through cosmological
birefringence, although the signals will be small even in the best-case
scenarios. Measurement of these bispectra requires only modest modifications to
the usual data-analysis algorithms. They may be useful as a consistency test in
searches for the usual bispectrum and to search for surprises in the data.
