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.