The National SOCIALIST WORKERS Party in Germany was a left-wing group. The fact that they were competing with Marxists does not magically make them right-wing extremists.
Mussolini was a Collectivist. He ran a Socialist newspaper. He declared that Fascism was a Collectivist philosophy.
How about this classic quote: "All within the state, nothing outside the state, nothing against the state." -- Benito Mussolini
Does he sound like a Tea Partier to you? Do you think you can make the case that Mussolini and Hitler were somehow political opposites??
Socialists in the US liked Hitler well enough after Hitler signed the non-aggression pact with Stalin (they liked Stalin better, but American Socialists were ready to support them both as a Dynamic Duo). Well, after Hitler invaded Russia, socialists hated him. They didn't want to be on his team anymore. Therefore -- voila! -- he was a right-winger.
Look at the history of the Left -- they have failed for centuries and they continually change their name so that they can separate themselves from past failures. Socialist. Communist. Marxist. Progressive. Liberal. It goes on and on. The fact that Hitler was revealed as a back-stabbing, genocidal maniac did not make him a "non-socialist" (far from it!) but it certainly gave the Left a reason to play their classic "name game" -- Socialist? Hitler?? Oh, we don't like him anymore. So, let's call him right-wing. That way everyone will be real clear that Socialists are the good guys. Even though Stalin and Mao killed more people than Hitler. He's bad. He attacked Uncle Joe, so he's not a Leftist. No sirree!
Politics is bifurcated -- Collectivism and Individualism. Right-wingers support individual rights, property rights, and less government intrusion. The other guys don't. Where do you think Hitler fits?
He than noticed that many Italian communists were disillusioned with communism's international focus, just as patriotism for the nascent Italian nation-state was growing.
European parliamentary politics created many unlikely coalitions, but fascism and NAZIism were merely socialism/communism with a nationalistic bent.
Both deny the rights of the individual (natural rights, not the phony, made-up "rights" commies say they are trying to get for the "people.")
They all assuage the collectivist impulse to control those who do things you don't like.
The left/right wing definition goes back to the French Revolution. Those who supported the ruling classes (aristocracy and old landowning families) were the right wing, those who wanted to overthrow them were the left wing.
Communists drew their power from labor unions, peasant uprisings, etc. Fascists opposed these and represented the interests of the ruling classes - the aristocracy, the industrialists, against such uprisings. The Communists were all about class warfare by the workers against the ruling classes, the Fascists and Nazis came to power by promising to put an end to such class warfare. Krupp and Porsche weren't about to back a movement that would disenfranchise them.
Saying that Fascists weren't right wing because they don't sound like the Tea Party is an absurd attempt to retroject issues of our time and place into another. It makes the same sense as asking whether Alexander the Great was a Democrat or a Republican.
The NSDAP began as a socialist and nationalist movement when it was founded by Anton Dexler. In order to win the support of the aristocracy, the bankers, the military, and the industrialists, Hitler eliminated the economic radicals from his party during the Night of Long Knives. With people like Roehm and Strasser out of the picture, the NSDAP was "socialist" in name only. No property (other than Jewish property) was nationalized, the banks stayed in private hands, and the aristocracy held on to their land.
Similarly, the fact that Mussolini began as a communist doesn't prove much about your point either, plenty of the most militant anti-communists started out as leftists in their youth.