I don't remember the Rumsfeld-Hussein handshake making the rounds, and that apparently was before I mostly stopped paying attention to political news, but my memories of how often this phenomenon occurs may not be representative.
If the hypothesis I proposed in the grandparent were correct regardless, then I imagine the significance behind the Rumsfeld-Hussein picture making the rounds is that the U.S. did back Hussein, so the upset was not that a member of our administration appeared to have associated with a negative figure, but that at one turn we built up a dictator when it seemed politically advantageous, and at another framed him as a monster who was worth taking down even if he didn't pose an active danger.
For all that my own policies tend to lean liberal though, I wouldn't credit the average Democrat with being especially discriminating; it's entirely likely that my above hypothesis is simply wrong.
Another one for the memory-is-really-unreliable file. Some researchers at UC Irvine (one of them is Elizabeth Loftus, whose name I've seen attached to other fake-memory studies) asked about 5000 subjects about their recollection of four political events. One of the political events never actually happened. About half the subjects said they remembered the fake event. Subjects were more likely to pseudo-remember events congruent with their political preferences (e.g., Bush or Obama doing something embarrassing).
Link to papers.ssrn.com (paper is freely downloadable).
The subjects were recruited from the readership of Slate, which unsurprisingly means they aren't a very representative sample of the US population (never mind the rest of the world). In particular, about 5% identified as conservative and about 60% as progressive.
Each real event was remembered by 90-98% of subjects. Self-identified conservatives remembered the real events a little less well. Self-identified progressives were much more likely to "remember" a fake event in which G W Bush took a vacation in Texas while Hurricane Katrina was devastating New Orleans. Self-identified conservatives were somewhat more likely to "remember" a fake event in which Barack Obama shook the hand of Mahmoud Ahmedinejad.
About half of the subjects who "remembered" fake events were unable to identify the fake event correctly when they were told that one of the events in the study was fake.