You're absolutely right, the hate community straddles both right and left, since the principles uniting them characterize neither. The test should be acceptance. From a political perspective, support from the hate community shouldn't be tolerated because of agreement on a few core issues, whether right or left. On that test I'd say the left is vurnerable. Haters aren't tolerated by the GOP, and rarely by credible political websites.
True. Evidence of this is that you don’t see neo-Nazis at, say, the Tea Parties, but you do see Communists and pro-palestinian anti-semites at virtually any large gathering of leftists. The Left is much more comfortable with its fringe than we are with ours.