If you're in an independent party that runs to the left of the Democratic party, you can usually depend on some sort of Republican support in an election campaign. That support is purely self-interested, of course; those Republicans hope to split the anti-Republican vote and win the election. But out of self-interest they may help you get on the ballot and participate in debates. The Democrats, in turn, will call you dupes of the Republicans while criticizing you along the usual lesser-evil lines, but if you really believe in your own cause and the necessity of your views, not theirs, prevailing, you won't care what they say.
Some ostensibly independent parties are actually less interested in winning office for themselves than in gaining power in the form of influence. These are the parties that practice cross-endorsement, where that's allowed. Such independents, if on the left, will offer their party line to Democrats who promise to prod their own party in a more progressive direction. These independents may also hope that the possibility of their running an actual independent candidate will scare Democrats into a more progressive direction. Cross-endorsers can expect less support from across the ideological aisle; the best such a group on the left can expect from Republicans is neglect.
The worst case scenario for a cross-endorsing party is to see the party it hopes to influence fleeing from its offer of aid, while at the same time the other big party tries to tie it around the neck of its rival like the proverbial albatross. That moment has come for the Working Families Party of New York. Beset with legal problems, it finds the Democratic gubernatorial candidate unreceptive to its enticements. Worse, Republicans have begun to treat the WFP as a liability for those Democrats who have accepted its endorsement. The GOP is calling on Democrats to repudiate Working Families in light of its legal peril. Its usefulness to the Bipolarchy it chose to play with rather than fight appears to be at an end. Its usefulness to the rest of us was probably never very great. A real independent party would never have found itself in such a predicament. Independence itself is a repudiation of the two-party system. If the system needs to repudiate an independent party, something was wrong from the start.