While that is the pop culture explanation it isn't quite right. In order for the parties to have flipped. The rural xenophobic bigots would have had to go to the Republican party. And the Republicans fascists would have had to have gone to the Democratic party.

The KKK and the bigots switched. The parties didn't really. The true irony being the KKK and Southern bigots were the huge influence on European fascists. But in the United States they were technically opposed simply because they had traditionally been in different political parties. But civil rights legislation brought about a realignment. Our ignorant xenophobic bigots and our fascist suddenly found themselves Brothers in Arms.

The party's polarized along the xenophobia and bigotry lines. But they didn't really switch.