Is there circular dichroism in rotating media? Does left-hand circularly and right-hand circularly polarized light get damped differently in a rotating medium, if the beams are travelling parallel to the curl of the flow (or the axis of rotation).
Even if the damping is the same for both directions the circular polarization gets affected according to

On the dragging of the plane of polarization of light propagating in a rotating medium
Proc. R. Soc. London, Ser. A349, 423(1976),
Author: M. A. Player

and

Miles Padgett, Graeme Whyte, John Girkin, Amanda Wright, Les Allen, Patrik Öhberg, and Stephen M. Barnett, "Polarization and image rotation induced by a rotating dielectric rod: an optical angular momentum interpretation," Opt. Lett. 31, 2205-2207 (2006)

 A: This is perfectly plausible.
One specific mechanism, proposed in this paper, is that left- and right-handed circularly polarized beams experience angular Doppler shifts in opposite directions when analyzed in the non-inertial frame of the rotating medium. If the absorption spectrum is non-uniform (and particularly in the vicinity of a hard absorption edge) then those rotation-induced frequency shifts can be sufficient to induce a nonzero circular dichroism.
That said, despite the concept being out there in the literature for 40+ years, I don't see any indications (at least from a quick search) of any experimental verification of this effect. This is probably attributable to a very high experimental difficulty: the frequency shift will be proportional to the mechanical angular velocity (which will rarely even approach the MHz range), and it is rare to find materials with optical absorption spectra that have such sharp edges. It's not necessarily an infeasible experiment, but it's also not necessarily easy and the payoff is likely not high enough for most groups to incur the expense.
As an additional note, one detail to consider is whether the rotation of the medium occurs globally (i.e. the whole sample is rotating rigidly) or at a microscopic level. The latter can be achieved in gas phase using an optical centrifuge, which can induce extremely high rates of rotation for each individual molecule about its centre of mass. In this case, I don't see any specific applications to circular dichroism as you've defined it, but there's plenty of work and some of it (e.g. this paper) does get close.
