  In two previous papers two evolution equations for the vortex line density
$L$, proposed by Vinen, were generalized to rotating superfluid turbulence and
compared with each other. Here, the already generalized alternative Vinen
equation is extended to the case in which counterflow and rotation are not
collinear. Then, the obtained equation is considered from the viewpoint of
non-equilibrium thermodynamics. According with this formalism, the
compatibility between this evolution equation for $L$ and that one for the
velocity of the superfluid component is studied. The compatibility condition
requires the presence of a new term dependent on the anisotropy of the tangle,
which indicates how the friction force depends on the rotation rate.
