Consider a heavy macroscopic object moving in a gas. Friction causes its kinetic energy to be converted into heat. Thermodynamically, there is (effectively) no entropy associated with the kinetic energy because all the energy is concentrated in a single degree of freedom. Therfore, if an amount $J$ of energy is converted from kinetic energy into heat, the total entropy change is $J/T$, so we can see that this is a spontaneous process.
But now consider an object moving relative to a gas with negative temperature. Such a thing has been created in the laboratory, so this is not just idle theoretical speculation. If an amount $J$ of kinetic energy gets converted into heat, the total entropy change is still $J/T$, but now this is negative. This seems to mean that the opposite process - conversion of heat into kinetic energy, accelerating the object - would be spontaneous.
This generalises to all other processes that convert work into heat. For example, performing the Joule heating experiment with a negative-temperature gas should cause the paddle to turn, and negative-temperature gas flowing through a pipe should experience an accelerating force rather than a decelerating one. Just as superfluids have zero viscosity, it seems that negative-temperature fluids must have negative viscosity.
I realise that this does not lead to perpetual motion. As heat is converted into work the inverse temperature ($1/T$) will increase until it reaches zero. But what does look odd is that in some ways the arrow of time appears to be reversed.
I realise that experimentally we're very far from being able to produce the macroscopic quantities of negative-temperature fluids that would be required in order to observe these things. But is it possible in principle? And if it is, would we actually see the phenomena I described, or is there some fundamental reason why they wouldn't happen after all? And has such a connection between negative temperatures and the arrow of time been discussed or debated in the literature?