  Generation, statistically steady state, and temporal decay of axially
rotating thermal counterflow of superfluid $^4$He (He~II) in a square channel
is probed using the second sound attenuation technique, measuring the density
of quantized vortex lines. The array of rectilinear quantized vortices created
by rotation strongly affects the development of quantum turbulence. At
relatively slow angular velocities, the type of instability responsible for the
destruction of the laminar counterflow qualitatively changes: the growth of
seed vortex loops pinned on the channel wall becomes gradually replaced by the
growth due to Donnelly-Glaberson instability, which leads to rapid growth of
helical Kelvin waves on vortices parallel with applied counterflow. The initial
transient growth of vortex line density that follows the sudden start of the
counterflow appears self-similar, linear in dimensionless time, $\Omega t$. We
show numerically that Kelvin waves of sufficiently strong amplitude reorient
the vortices into more flattened shapes, which grow similarly to a free vortex
ring. The observed steady state vortex line density at sufficiently high
counterflow velocity and its early temporal decay after the counterflow is
switched off is not appreciably affected by rotation. It is striking, however,
that although the steady state of rotating counterflow is very different from
rotating classical grid-generated turbulence, the late temporal decay of both
displays similar features: the decay exponent decreases with the rotation rate
$\Omega$ from -3/2 towards approximately -0.7, typical for two-dimensional
turbulence, consistent with the transition to bidirectional cascade.
