According to Roger L. Simon, the Republicans and Democrats have nicknames for each other: the dumb party and the evil party. Can you guess which is which? Now go to his site and have a look.
This sort of thing has a long history. The liberal and conservative factions in Mexico's endless civil wars used to be the 'thieves' and 'drunks' respectively, which sounds about right, though you could probably have reversed the epithets. But (without any patriotic bias) the most interesting party names are British, i.e. Whig and Tory. A Whig was originally a sort of Scottish bandit/rebel: the name was intended to smear the anti-monarchist party by association with Scotland and consequently the Presbyterians. Similarly a Tory (Gaelic toridhe) was a sort of Irish bandit/rebel, the intended smear in this case being that the monarchists were closet Catholics. We don't know for sure who started it, but we know when: around 1680, at the time of the Exclusion Crisis.
Could anyone out there think of some good further examples?