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Disk galaxies can exchange angular momentum and baryons with their host darkmatter (DM) halos. These halos possess internal spin, `lambda', which isinsignificant rotationally but does affect interactions between the baryonicand DM components. While statistics of prograde and retrograde spinning halosin galaxies is not available at present, the existence of such halos isimportant for galaxy evolution. In the previous works, we analyzed dynamicaland secular evolution of stellar bars in prograde spinning halos and the DMresponse to the bar perturbation, and found that it is modified by the resonantinteractions between the bar and the DM halo orbits. In the present work, wefollow the evolution of stellar bars in retrograde halos. We find, that thisevolution differs substantially from evolution in rigid unresponsive halos,discussed in the literature. First, we confirm that the bar instability isdelayed progressively along the retrograde `lambda' sequence. Second, the barevolution in the retrograde halos differs also from that in the prograde halos,in that the bars continue to grow substantially over the simulation time of 10Gyr. The DM response is also substantially weaker compared to this response inthe prograde halos. Third, using orbital spectral analysis of the DM orbitalstructure, we find a phenomenon we call the orbit reversal --- when retrogradeDM orbits interact with the stellar bar, reverse their streaming andprecession, and become prograde. This process dominates the inner halo regionadjacent to the bar and allows these orbits to be trapped by the bar, thusincreasing efficiency of angular momentum transfer by the Inner LindbladResonance. We demonstrate this reversal process explicitly in a number ofexamples.

Stellar Bars in Counter-Rotating Dark Matter Halos: The Role of Halo Orbit Reversals