"Kalterkrieg"= a Cold War between a somehow victorious Nazi Germany and the US rather than a victorious USSR and the US.
Of course, this depends on exactly how such a thing happened. Assume in this case, the end result is Nazi Germany is roughly in the same place vis a vis the US as the USSR was in real life. (Including--getting nuclear bombs real quick after the war.) I'm thinking more about US culture in an alternate Cold War with a different enemy, not in another alternate WWII.
Some random thoughts follow.
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The Pledge of Allegience would be rewritten differently. I imagine the last bit would go something like "...one nation, indivisible, with liberty and justice for all, regardless of race, color, or creed." To emphasize US ideological differences from Naziism rather than from Communism.
Most likely, liberals would be more enthusiastic about mandatory Pledging than OTL, and Southern whites less enthusiastic. Those who completely refused in the *1950s (including groups like the Jehovah's Witnesses who refuse to take oaths on religious grounds) would be accused of being Nazi sympathizers rather than Commies; this changes how well the charge resonates with different groups of mainstream Americans.
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More emphasis on the civil rights movement is likely; the US is distancing itself from hostile Nazis, not hostile Communists. This isn't all positive--the amount of pushback is going to be greater.
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People have disagreed with me on this before, but I still say that some similarities between TTL and OTL, namely facing a hostile ideology headquartered in a hostile superpower, would mean some similar reactions. Namely, the 1950s will be a period of frightened conformity, with efforts to root out suspected agents of that ideology/superpower--a "witch hunt", even. (Perhaps even led by Senator McCarthy, jumping on the alternate opportunity.)
But the nature of the adversary means some differences, too. First, Hollywood will be firmly on board; there were still Communist Party members in the Hollywood establishment in the 50s, but few if any Nazis. Second, Nazi intelligence never penetrated the US like Soviet intelligence did; there's going to be fewer actual "witches" out there. (I don't know if that would make the "witch hunt" more or less severe, though). Thirdly,if Southerners are targets of the hunt rather than hunters, this could get ugly; they have more numbers and regional power to retaliate than the American Communist Party ever did.
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Expect J. Edgar Hoover to be fondly remembered on the Left; most of the FBI's efforts will be directed at the KKK and the like. The FBI may even get away with a bigger carte blanche than they had in OTL 50s.
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Either outlaw biker gangs adopt hammers and sickles rather than swastikas and iron crosses to flaunt their bad boy credentials; or they get stomped harder, seeing as how they can be reasonably suspected of declaring allegience to the hostile superpower.