  Almost all studies of vortex states in helium II have been concerned with
either ordered vortex lattices or disordered vortex tangles. This work studies
numerically what happens in the presence of both rotation (which induces order)
and thermal counterflow (which induces disorder). We find a new statistically
steady state in which the vortex tangle is polarized along the rotational axis.
Our results are used to interpret an instability which was discovered
experimentally by Swanson et al. years ago but has been unexplained until now.
