Labels of "left" and "right" and even "liberal" and "conservative" are often confusing and the meaning of the terms can change over time. Classical Liberalism would now be called conservatism. When the Soviet Union was collapsing those who wanted to maintain a Communist empire were called "conservative." Nazis are considered part of the right but Nazis were National Socialist who opposed free market capitalism and believed that people should subordinate their personal interests to the common good.
In an article in the Tennessean, "White Lives Matter rally: Who are the groups involved, and what do they believe?", the believes of the principle participants in the march were laid out. The bigots organizing the Murfressboro and Shelbyville protest are left wingers when it comes to most issues other than race.
The Traditionalist Workers Party opposes capitalism and colonialism and calls for a “National Socialist government, economy and society for our people.” There website echos Occupy Wall Street saying, "Democrats and Republicans put Wall Street Greed Ahead of Main Street Values." They also demand an "end of foreclosures and taxation on the primary dwelling places." On environmental issues they say, "We stand for investing in and developing renewable and green energy sources for long-term energy independence and reduction of carbon emissions to give clean affordable power to our nation, millions of jobs to our citizens, and cleaner air and water for our families." On healthcare they say, "Healthcare is a right for our people, not a business opportunity where working families and the poor are left without adequate access to doctors, medicines and proper treatment."
The National Socialist Movement calls for a livable wage, the ending of taxes on food, medicine, housing, clothing and other necessities, affordable housing, and universal health care. They "demand the nationalization of all businesses which have been formed into corporations." They are also strong environmentalist. They see access to higher educations as a right. They advocate for "the common good above self interest."
Vanguard America is a group that has splintered several times and recently Word Press closed down the organization's website so one can not get first hand information about what they believe but the group's "manifesto" does not reveal many positions on issues other than those related to race. They do condemn global finance and say of ​multinational corporations​ they "have bled this nation dry will not be allowed to continue their detrimental efforts unabated." The League of the South does not address any issues except those related to their advocacy of session from the union.
When one hears white supremacist groups defined as "right-wing" or "alt-right" keep in mind that many of them are anti-capitalist, favor a system of national health care, and are environmentalist. They could just as easily be called socialist or leftist.