I firmly believe that the left and right wing concepts are purely economic and unrelated to the severity of control the government plays in politics. I believe they are two separate concepts because there can be non-fascist communism and non-fascist capitalism.

Modernized political scales throw the wings into economic handelings while amount of government control is on a scale of Libertarian - Authoritarian.

Fascism, historically, is towards the middle since they do have a centralized bank and centralized health-care but they still allow for people to start their own businesses and trade somewhat freely (so long as the people aren't deemed inferior for elitist reasons.)

There's also a lot of businesses being told to manufacture for government, specifically the military industrial complex.

State Communism (seen under Stalin and Mao) are Left-wing Authoritarian systems, where the government runs most of the traditionally market duties. There is no or little private sector.

The issue with having a truly Right Wing Authoritarian system is that Authoritarianism that doesn't touch the market at all is rare. At the same time, extremely left wing economics tries to put the economic sector into the collective hand, not the hands of the state. That requires a different level of action, one that many anarcho-socialists try to persuade the people by.