“After all, they believe that only the right is extremist and violent.
Not quite. They believe that violence and extremism is a virtue, but only when their fellow travelers engage in it.
Neo-Nazis ARE members of a left-wing organization.
There is nothing “right-wing” about National Socialism. Only in relation to Communism was the doctrine of the “National Socialist German Workingmen’s Party” even remotely to the right. Their methods and tactics were virtually indistinguishable from those of the regime of Joseph Stalin, except they kept much better records.
“Command and control” regimes all resemble each other in more way than they differ, and it all comes down to stifling individual initiative, onerous taxation, and regulation of even the minute aspects of daily life.
Backed up with vast prison complexes and armed might exerted against unarmed subjects.