The Bavarian Illuminati are (rather ironically) an example of an actual political conspiracy whose beliefs would be pretty unremarkable today. They were liberal humanists; they believed in freedom of religion, reason, improving people's morals by studying secular ethics, and republican government. Why were they secretive and conspiratorial? Because they were operating in 18th-century Bavaria, a conservative Catholic monarchy where religious dissent was illegal, the secret police investigated social groups to uncover political dissent, and republicanism would mean overthrowing the government.

It's a silly counterfactual, but I can't resist imagining that if Weishaupt lived today, he'd post on Less Wrong.