The real reason is that so long as the Soviet Union existed, “right wing” had the coherent meaning “opposed to the Comintern”, while “left wing” meant supporting the goals of the Comintern. Thus Nazis and Fascists whose economic and social programs were largely identical to Stalin’s, along with monarchists, clericalists (of whatever Christian confession), Tories, free marketeers of the sort that used to be called “liberal” who are now, in America, called conservatives, were all “right wing”, while “fellow-travelers” (liberals in current usage) along with Communists who toed the Party line out of Moscow were “left wing”.
(Yes, yes, I know they originally had to do with seating the French parliament at the time of the Revolution, but that meaning is long gone.)
The usage stuck because the Marcusian multiculturalist left has the same opponents the Comintern did, even though for practical reasons (socialism and communism don’t work, while fascism does) they have all become fascists in terms of their economic program.
I wouldn’t say the French revolution meaning is gone. People still tend to think in terms of left = radical, right = reactionary, and center = status quot. Or perhaps better put: center = keep things as they are, left = change things, right = put things back the way they were. If you get any more specific than that, for instance making the left more democratic and the right more aristocratic, it soon loses all meaning. But in a very basic sense the old meaning still holds.
Unfortunately, several new meanings have been superimposed on the basic one, for instance commies being to the left and nazis to the right. One of the biggest defects of the classical spectrum is that it can’t possibly sustain such a polarization. It would be fortunate, then, if the old French revolutionary meaning had been lost by now. But since it hasn’t, people are driven to the absurdity of mistaking nazis for being oldfashioned.