- Historically, fascism came from socialism
- The 3rd Position: their core economic philosophy was moderate democratic socialism and temporarily allowing private business (as proxies for the state), as a stepping stone on the path to pure statism (Socialism): "First Brown then Red".
- Fascist leaders were all Socialists
- Their symbol (and the word) is a bundle of sticks tied together, often with an axe in the middle representing "all of us, are stronger than one of us" and collective power . Meaning collectivism over individualism.
- Their core belief is in the state (collective) above individuals (and individual liberties)
- Fascists despised the status quo and considered themselves progressive (forward looking), and were not attracted by a return to bygone eras -- they were the opposite of conservatism or right wing movements
- The progressive left in America supported the fascists, and the fascists borrowed their ideas and celebrated them back (until the war)
- National Socialism, was the German branch of Fascism, that added in cultural components of Germany (anti-semitism and more imperialism).
- and were.
Since modern leftist movements don't like their ties to fascism, they throw out a lot of fallacies and distraction to obfuscate that truth, like:
- Since Fascists and other socialists fought, they can't be socialists
- Since they were militant, imperialistic, nationalistic and racists, they couldn't be progressives/socialist (Germany far more than Italy)
- Since they were authoritarian, and Socialism is democratic, they weren't socialists.
- The big lie: the other side is more like fascists than we are!
Each of those arguments is easily debunked below.
History: Fascism is socialism
However, Marx's Communism failed to deliver on its promises:
- According to Marx, the natural progression was Feudalism -> Capitalism -> Communism.
- Marx felt that Capitalism was flawed as all resources would pool with the few, thus the distance between worker and 1% would grow (as would resentment), therefor the most capitalist countries (Germany, U.S., England) would fall first (and soon). It didn't happen. Instead, things had gotten better for developed countries workers and they increased stability over time.
- It was non-capitalist and poorly industrialized Russia that collapsed into communism (starting in 1906, and ending in 1917).
- And it didn't happen by the workers spontaneous rising up (as promised) but by elites in Universities and elsewhere driving the change.
- Plus, Marxism's violence, despotism, economic ruin and famines, wasn't at all what was promised or other Countries wanted to mimic.
Marxist theory was broken (as to why/how), so the Socialists needed new theories to be invented to justify their beliefs.
Marxist Communist-Socialism fractured into two schools, Leninists and Fascists. Both were revolutionary socialists (Marxists). Both accepted that Marx had been wrong, and that spontaneous workers revolutions (what they promised and desperately wanted) wouldn't just happen on its own, in stable developed countries.
- Leninists felt that by exploiting foreign workers, Capitalism was still thriving on the backs of workers -- but because it was foreign workers, they had bought off the domestic ones (or underclasses). Thus outside forces were necessary to drive Communism to the less enlightened.
- Fascists, knew that revolution wouldn't work in stable developed countries, so they needed a more incremental and pragmatic approach to Socialism: one where you'd transition from private owned business and personal property to state owned, through an intermediate step -- where the state controlled private business (the means of production) and private property through proxies. The owners were TEMPORARILY allowed to keep their businesses (and avoid nationalization), and people could keep private property, but only as long as you were doing what the state wanted. But once they taxed, regulated and lawed their way past a certain point, they could finish the transition to Socialism. The intermediate step was called the 3rd Position.
- Socialist Academics contributed to the rise of the 3rd Reich by supporting this ideology. 
The 3rd Position
Fascisms view of individuals and property rights:
- Communism/Socialism is the idea that you have no property rights, (only the state does: e.g. everything is "communal"), and thus it is the states job to redistribute it's property (wealth) fairly amongst the people
- Capitalism (in the extreme) is the idea that only individuals have property rights. You can pool those rights briefly with government or corporate agencies (and let them be stewards of that private property). But property as a concept, is reserved for individuals.
- Fascism was the third position 
Thus everyone was part of the collective. You could have corporations, businesses and private property, but only if they were putting the states interests first (as defined by the political class). Mussolini described it as a "merger of state and corporate power", over the individual (anti-libertarianism). (The same rhetoric and position as the Occupy Movement). Today we'd call it Crony Capitalism, or a regulatory state -- private business being the proxies of government, along with government subsidies to support them (Solar Power, Electric Cars, subsidies like that).
So Capitalism is an individualist (libertarian) ideology, while Communism, Socialism and Fascism are all collectivist (authoritarian) ideologies. As long as you put the states interests ahead of your individual ones, you can exist. As soon as you're in conflict with the state/collectives interests, they can nationalize. You have the freedom to do what the state wants, or your freedom will be taken away. Which isn't really freedom at all.
When you look into any of the leaders or voices in the fascist movement, they were all Socialists. They varied from Marxists, Leninists, or branches and factions -- but they were a pro-union, workers movements, whose rhetoric was virtually indistinguishable from other socialists, unless you were deep in the weeds of how to govern to achieve socialist nirvana.
Here's some key names:
- Giovanni Gentile - The "philosopher of Fascism", who ghost wrote Benito Mussolini's, "The Political and Social Doctrine of Fascism”, as well as many other books on the topic. He was a follower of Hegel and Marx, member of the Fascist Grand Council in 1925, and the last president of the Royal Academy of Italy. A devout Socialist and believer that the state belonged ahead of the individual.
- Corrado Gini - The guy who wrote the book on economic fascism and invented the Gini Coefficient believed in the state above individuals, was a proponent of Eugenics (popular among the statist left). 
- Benito Mussolini - a lifelong Socialist. While he hand limited power in Italy (due to governmental structure), when he was made President in Exile, he immediately nationalized the businesses, privatized the wealth, and enacted every Socialist reform possible. The only reason he hadn't done that in Italy before then, was because of the limitations of power.
- Adolf Hitler - he never called himself or his party Fascists (though he attributed a lot to them), he was always a National Socialist, and all his plans were to enact socialism, just incrementally: first win the War with the help of private business, then Socialism.
For if the nineteenth century was a century of individualism (Classical liberalism always signifying individualism) it may be expected that this will be a century of collectivism, and hence the century of the State. —Benito Mussolini, "The Political and Social Doctrine of Fascism,” Jane Soames authorized translation, Hogarth Press, London, 1933, p. 20. Against individualism, the Fascist conception is for the State; and it is for the individual in so far as he coincides with the State . . . . It is opposed to classical Liberalism . . . . Liberalism denied the State in the interests of the particular individual; Fascism reaffirms the State as the true reality of the individual. (p. 13) 1935 version
was this hybrid Socialist system that believed in still allowing private business, as long as it was doing what the authoritarian state dictated. (High regulation, taxes, and central control).
- They fundamentally believed that Socialism (state planning and control) lead to better organization, output, and was more advanced than western liberalism (individualism), this is often what their long speeches and ceremonies were about: the superiority of their beliefs in their collective, and attacking anyone outside it as inferior (whether based on nation, race, religion, or political beliefs, everyone else was inferior, and needed their wisdom forced upon them, for their own good). (This ties in with the Law of Jante).
- Fascisms roots, origins and leadership, all came from the left: Socialist, pro-worker, populist, secular-progressive intellectuals, haters of existing society (and especially of its most bourgeois aspects: "eat the rich"), nationalistic tendencies of anti-immigrant/anti-competition labor, and they frequently used pseudo-science to rationalize their quest for power (Eugenics, Gini Coefficient, belief in top-down command-economies being governed by a "brain trust", and so on).
When Marx's Communism (revolutionary socialism) failed to deliver on it's promises, the far left ideologies started fracturing: and one branch was this hybrid Socialist system that believed in still allowing private business, as long as it was doing what the authoritarian state dictated. (High regulation, taxes, and central control).
They fundamentally believed that Socialism (state planning and control) lead to better organization, output, and was more advanced than western liberalism (individualism), this is often what their long speeches and ceremonies were about: the superiority of their beliefs in their collective, and attacking anyone outside it as inferior (whether based on nation, race, religion, or political beliefs, everyone else was inferior, and needed their wisdom forced upon them, for their own good). (This ties in with the Law of Jante).
National Socialism (Nazi's)
- National Socialism, was a branch of fascism that added in cultural components of Germany (the anti-semitism) and came specifically from a branch of socialism called National syndicalism. (A syndicate is a group of workers/unions unifying an industry, and these groups of syndicates/unions came together to unite the nation).  The leadership and followers had branched off and became National Socialists German Workers' Party to better describe their philosophy of collective socialism and nationalistic purity. 
- Those that call it a "right wing" movement are either intentionally misleading (by being pedantic in a way that they know will be taken wrong), or are blatantly ignorant of what the terms mean or the history of the movement.
Fascists and Socialists fought
The Socialists/Communists supported Fascism with the cry, "First Brown, then Red"... until the fascists went on to purge those that weren't politially-correct enough to be fascists (including vocal Socialists/Communists). But this is a common in Socialism. They all agree in loose concept to go to socialism... but they disagree on what the final form looks like, who should be in charge, and how to get there. So infighting between the branches and implementations of statism is the norm. They can only work together long enough to destroy the non-statists, then they fight for which branch of statism should be in control. So the the example of fighting between Fascism and Communism (or Socialism) is not evidence they were on opposite sides at all, but this was much more like Hillary Clinton and Bernie Sanders fighting over who better represented the far left view for America.
The Communists of the time, returned the favor or persecution with their Antifa (antifascist) movement. (Both sides were revolutionary radicals that were willing to use violence against the other side).
While in America, our terms reversed: individualists (libertarians, classical liberals, conservatives) tend to pool on the right, not the left, so the terms/meanings/roles are directly reversed. If Fascism is right wing in Europe, it's left wing in America (or they were using a different dimension to compare it on. The same way in America, Conservatism means go backwards: back when we had less government control, but in many European countries, conservatism can mean going back to when they had Monarchs and more authoritarian control. These terms don't translate as well as some people think.
There's a lot of other ways to look at right versus left wing: none of them show fascism to be a purely right wing belief system, while all show it to be a strongly left wing one (by American definitions).
Something wicked, this way comes
- Fasces: https://en.wikipedia.org/wiki/Fasces
- Socialist Academics contributed to the rise of the 3rd Reich: https://fee.org/articles/socialist-academics-contributed-to-the-rise-of-the-third-reich/
- : that business (and the individuals) could have property rights, but only as proxies for the state (if they were doing what the state decided was in the public/national good).
- Corrado Gini:
- National Syndicalism: https://en.wikipedia.org/wiki/National_syndicalism
- National Socialists: https://en.wikipedia.org/wiki/Nazi_Party
- Intentional Bias or Incompetence? Daily Caller gives Googlers enough credit that they think it's intentional. I suspect there's just enough college-miseducated herd-following millennial working there, that they sincerely don't know better: http://dailycaller.com/2017/02/04/google-redefines-the-word-fascism-to-smear-conservatives-protect-liberal-rioters/