Conservatives are really middle of the road in the political spectrum. as you move left on the spectrum your get extreme left, then socialists, then communists. As you move right your get extreme right, and then moving to the extreme extreme right would be the Anarchists.
The farther left you move the more they want the govt to control everything. As you move right they want less govt. Go all the way right to the anarchists and they want zero govt.
So us moderate conservatives are really the ones well balanced and that is why the Right is Right.
So this thinking, while understandable, is based on the
Traditional Left-Right Spectrum that emerged around the French Revolution. The Monarchists and Aristocrats sat on the Right side of the Assembly while the Revolutionists and Commoners sat on the Left side. The Fascism of Germany, Spain, and Italy generally align on the Right side because they all supported Monarchism/Imperialism to some extent. American political parties don't really fit into this context because neither the Left nor the Right associate with Monarchism or Revolution. Democrats are essentially Social Liberals/Egalitarian Liberals, while Republicans are Classical Liberals.
A modern, single plane, left to right US political chart is essentially useless when discussing things like this. Even your definition of Anarchy is only in the American, anarcho-capitalist sense (that emerged as an offshoot of American libertarianism), since anarchism has traditionally been a "left" wing
anti-capitalist state, the most famous being the Spanish ararchist settlements that were killed by Franco's forces in the 30's. Although Anarchy does have many schools of thought.
The Fascists themselves rejected the labels of Left and Right, and instead called themselves the Third Position, or Third Way, that was outside of the Left-Right Spectrum. It was both anti-Communist and anti-Capitalist. It rejected the Materialist worldview of the Marxists and also the Liberal worldview of the Capitalists. They instead embraced
Corporatism and an
Actualist worldview. This was basically a system where massive labor unions and massive organizations of capitalists managed the economy, but not labor unions in the small American sense, a labor union in Italy would have been called something like the transportation union(apologies that I cannot remember the exact term off the top of my head) with
all jobs related to transportation under that umbrella, bus and train drivers, dockworkers, road construction workers, etc. Capitalist groups in reference to Italy would in modern terms be something similar to the entire combined tech or oil industries we have. Like if Google, Apple, Amazon, Cisco, Level3, Facebook, Microsoft, all united under one umbrella.
To elaborate a bit, Corporatism is not Socialism, but it comes close to the positions many European Social Democrats had taken during and after WWI. From this perspective, Fascism was also close to the moderate Left in Europe but deeply opposed to the extreme Left and Bolshevism/Marxism.
With both of these points in mind: Yes, there were Left-wing Fascists within the Fascist movements but the overall Fascist movements were not strictly left-wing. So no, there were no left-Fascist governments.
The biggest point to make is that Fascism is not
strictly Right or
strictly Left. There were such things as Homosexual Nazis, Socialist Fascists, anarcho-Corporatists, Actualist-Communists, and so on, that existed within Fascist governments or within Fascist movements. Each one of these groups exerted some degree of influence. But many of these groups were often also purged after the party had solidified itself into power.
We tend to want to group things into categories we understand, but when you are looking at historical things like the rise of fascism, you have to try to understand that those arose from sometimes hundreds of years of unique political and societal changes that we have no innate historical knowledge of growing up. Like as an example, could someone just say the entire rise of the Soviet Union was because "Lenin hated people with money"? You could if you want. But there is literally hundreds of years of history, politics, and social shifts spanning an insane amount of readings that led to the October Revolution. Entire volumes of books have been written about it. Same with Italy and Nazi Germany.
Sorry about the wall of text, but political theory and historical contexts are kind of my thing, and I feel that we should have an appreciation of exactly how not-simple any of these massive political movements were.
I can't believe I forgot this, but in addition to the references in my above post,
This essay by Umberto Eco is usually a top resource when it comes to Fascism, with his 14 characteristics of Fascism widely regarded as its best descriptors.