For example, is it United Auto Workers or Autoworkers, United Farm Workers or Farmworkers, United Steel Workers or Steelworkers?
United Auto Workers
United Farm Workers
For a healthy serving of schadenfreude, troll Google News for botched references to United Autoworkers, United Farmworkers and United Steel Workers. Then see how many articles by those paid to know better refer erroneously to the Communication Workers of America (correct: Communications), International Longshoreman's Association (correct: Longshoremen's) and Laborer's International Union of North America (correct: Laborers').
Mr. Rewrite is a big fan of this World Wide Web thing. One can easily confirm proper nouns through a "search engine." One also can be "old school" and consult the AP Stylebook entry on union names.
-- When dealing with union locals, check out the AP Stylebook entry on local of a union. It says to use the figure and a capital L: Local 123 of the American Brotherhood of Usage Nerds.
-- AP is fine with AFL-CIO in all references when referring to -- deep breath -- the American Federation of Labor and Congress of Industrial Organizations.
Mr. Rewrite had better shut down this dense post or even his most loyal readers will go on strike.