The NSM is just the old American Nazi Party. When the NSM organizes with the Traditional Workers Party, that is called the National Front. The head of the Traditional Worker Party is Matthew Heimbach and he's the guy that beat up the black woman at the Trump Rally, while he was wearing his red MAGA hat.
Other groups: League of the South, Vanguard America, National Policy Institute, Proud Boys, Council of Conservative Citizens(remember Trent Lott). Alt-Right.com, Daily Stormer, Identity Dixie, Identity Europa, Red Elephants, American Freedom Keepers, Pennsylvania Lightfoot Militia, Fraternal Order of Alt-Knights, White Knights of the KKK, East Coast Knights of the KKK, Anti-Communist Action(anti-Antifa)
Real Republicans have roundly rejected them. But the MSM never reports that!
As it was for the National Socialists in Germany - race. It's not about joining the Democrats or not being able to join the Democrats. The Democrats are also a party of racialism - a vacuous 'philosophy' of pitching races against races qua anything non-white and white self-loathing vs whites but this breaks down further in that blacks, Jews, Latinos, East Indians, etc. who are conservative are not really that ethnic/cultural/racial group because they have sold out to the conservative side.To the extent that race baiting and racialism is an essential core of the National Socialists/Nazis etc. it is also for the Democratic Party. The latter have simply turned the white notion on its head.
The National Socialists, David Dukes, etc. can't be comfortable with conservatives and Republicans (even though the latter has been bastardized the last many years regarding globalism) because racialists, but more particularly National Socialists, which this guy was a card carrying member, as they were in Germany congregate around race as pre-eminent but that could only be maintained in a socialist socio/political construct as the State is the supreme and organic vehicle to control both capital and social order. I'm assuming your reference to Matthew Heimbach and wearing a MAGA hat was a typical attempt as it is with communists/anarchists etc.to foment racial discord and perceptions by planting operatives to play racial propaganda games.
They always are aligned with the GOP. David Duke ran as a Republican. Augustus Invictus ran as a Republican.
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Duke first ran as a Democrat. Invictus first ran as a Libertarian.