In the immediate aftermath of 9/11, Putin's Russia made clear its support of America's "war on terror", not least because it gave the Russians an excuse to crack down on their Chechen 'problem' without facing too much international condemnation. Taking this a little further than OTL, what if the Russians had contributed troops to the multinational force that ejected the Taliban from Afghanistan - either during the 'war' phase, or later as some coalition partners did?
If Putin made the offer, I can see it being a difficult decision for the Americans - it would be valuable diplomatically in the West, but would send the wrong message in Afghanistan (after backing the mujahideen against the Russians in the 80s, it's now the other way around...).
If Russian troops served alongside U.S. and multinational forces in 2001/2, would this alter what came afterwards? Say Russia still opposes the Iraq war, but in a more measured way like France and Germany OTL, and continues to support operations in Afghanistan (like France and Germany). Would this avoid the recent cooling of relations between Moscow and Washington (and especially London) or would that still happen regardless?