  Interplanetary coronal mass ejections (ICMEs) contain magnetic field and
velocity fluctuations across a wide range of scales. These fluctuations may be
interpreted as Alfv\'enic wave packets propagating parallel or anti-parallel to
the background magnetic field, with the difference in power between
counter-propagating fluxes quantified by the cross helicity. We have determined
the cross helicity of inertial range fluctuations at $10^{-3}-10^{-2}$ Hz in
226 ICME flux ropes and 176 ICME sheaths observed by the Wind spacecraft at 1
au during 1995-2015. The flux ropes and sheaths had mean, normalised cross
helicities of 0.18 and 0.24, respectively, with positive values here indicating
net anti-sunward fluxes. While still tipped towards the anti-sunward direction
on average, fluxes in ICMEs tend to be more balanced than in the solar wind at
1 au, where the mean cross helicity is larger. Superposed epoch profiles show
cross helicity falling sharply in the sheath and reaching a minimum inside the
flux rope near the leading edge. More imbalanced, solar wind-like cross
helicity was found towards the trailing edge and laterally further from the
rope axis. The dependence of cross helicity on flux rope orientation and the
presence of an upstream shock are considered. Potential origins of the low
cross helicity in ICMEs at 1 au include balanced driving of the closed-loop
flux rope at the Sun and ICME-solar wind interactions in interplanetary space.
We propose that low cross helicity of fluctuations is added to the standard
list of ICME signatures.
