In 1944 George Orwell published 'What is Fascism?' He said:
It will be seen that, as used, the word ‘Fascism’ is almost entirely
meaningless. In conversation, of course, it is used even more wildly
than in print. I have heard it applied to farmers, shopkeepers, Social
Credit, corporal punishment, fox-hunting, bull-fighting, the 1922
Committee, the 1941 Committee, Kipling, Gandhi, Chiang Kai-Shek,
homosexuality, Priestley's broadcasts, Youth Hostels, astrology,
women, dogs and I do not know what else...
By ‘Fascism’ they mean, roughly speaking, something cruel,
unscrupulous, arrogant, obscurantist, anti-liberal and
anti-working-class. Except for the relatively small number of Fascist
sympathizers, almost any English person would accept ‘bully’ as a
synonym for ‘Fascist’. That is about as near to a definition as this
much-abused word has come.
But Fascism is also a political and economic system. Why, then, cannot
we have a clear and generally accepted definition of it? Alas! we
shall not get one — not yet, anyway. To say why would take too long,
but basically it is because it is impossible to define Fascism
satisfactorily without making admissions which neither the Fascists
themselves, nor the Conservatives, nor Socialists of any colour, are
willing to make. All one can do for the moment is to use the word with
a certain amount of circumspection and not, as is usually done,
degrade it to the level of a swearword.
It's clear that most people use the term to mean something they consider awful and extreme. Generally we could say a 'fascist' is anyone more right wing than you, just as a 'communist' is anyone more left wing than you. However, these intuitive and frankly bullshit explanations demonstrate a complete lack of ideological comprehension.
Fascism has been described as 'third way', incorporating aspects of what is traditionally considered 'left' and 'right' into something very different. Whatever way you attempt to generalise left and right, it tends to fall into some sort of a trap. To understand 'left' as being the pursuit of revolutionary collective equality, in opposition to 'right' being the pursuit of traditional individual freedom, does at least touch on the origins of the distinction from the French revolution. But this is clearly insufficient to describe fascism.
Complicating matters is the fact that there is considerable variation between fascist regimes; Italy, Germany, Spain. But there are common elements. Fascism experts often conclude fascism's uniqueness is in its belief in violence as morality. Fascism is anti-democratic, anti-individualist, anti-intellectual, and pro-violence for its own sake; not just as a means to an end.
Fascists believe that violence is a life-affirming act which allows the natural order to be restored. While the prioritisation of the collective above the individual may be considered typically left-wing, equally the emphasis on natural hierarchy and inequality is undeniably right-wing. So with Fascism we have an ideology which doesn't make any sense in a left-right dichotomy.
To the fascist, democracy is a perverse exercise which denies the strong the ability to rule over the weak. This requires violent revolution to conquer society, that this act will be proof of their superiority.
In the fascist state the people become a superior race through transcendent purging violence, in the same way as the body combats infection. This has led to interesting contradictions. Hitler was supremely anti-Semitic, regarding Jews as a fundamentally alien threat and global problem. While, initially at least, Mussolini was forced to concede that in Italy Jews were not alien others. Between seizing power in 1922 and 1938, Italy's Jews suffered little persecution. But as Italy's relationship with Germany intensified so too did the transfusion of German anti-Semitism.
World expert in the holocaust, Timothy Snyder, presents an even more counter-intuitive analysis of Hitler's ideology and Nazism more specifically. Snyder concludes that Hitler cannot be understood as merely a nationalist or anti-Semite who went further than most, but that his ideology was unique: 'racial anarchism'. What he means by this, is that Hitler viewed most things; religion, philosophy, rule of law, liberalism, communism, etc, as essentially inventions of a Jewish conspiracy to destroy superior races.
In this context, Nazi activity in occupied Poland makes a great deal of sense, as the Nazi regime was not simply occupying a foreign state, it was attempting to completely destroy all institutions which prevented racial warfare. What used to be the Polish state was reduced to a region of effective lawlessness in which the allegedly superior race could do what was natural, and wipe out the competition.
Anarchism has typically been defined as a far-left ideology which promotes the equality of individuals, and rejects the necessity of an involuntary state. Anarchists advocate for a 'free territory' of voluntary association and fair distribution of resources. This is quite the contrast to Nazi anarchist ideas, which are about the violent enforcement of inequality. So again we have a direct conflict within the traditional left-right dichotomy.
It has been said that because the Nazis were 'National Socialists' that they must therefore be socialists, but if you believe that perhaps you also believe the Democratic People's Republic of Korea has free and fair elections.
I would tend to accept self-identification, but this does require a basic ideological consistency. Catholics and Protestants both claim to be Christian, and they both adhere to a lot of common fundamental ideas about Jesus which does suggest they are both a sort of Christian. Not so with socialist beliefs about equality and fascist beliefs about inequality, there is no commonality.
The state control of industry is by no means uniquely socialist. After all, state corporations were part and parcel of imperialism which was neither socialist nor fascist; like the British East India Company. In capitalism industry exists for individual profit. In socialism industry exists to share surplus. In fascism industry exists to be loyal to the state; fascist states had varying economic policies and didn't seem to care much for consistency in this regard. There simply isn't a 'fascist' economic model.
In conclusion, fascism is third way because it does not make sense in a left-right dichotomy. The pursuit of revolutionary collective inequality doesn't fit anywhere on that spectrum.
EDIT: I watched a documentary 'Ben Building: Mussolini, Monuments and Modernism' which has one of the best distinctions I've yet heard:
What is fascism? The response that it's a pathology of the extreme
right is idle, thoughtless. Equally, the response that it's a cancer
of the extreme left is idle, thoughtless. If the extreme right is a
race horse, and the extreme left is a cart horse, what sort of horse
is fascism? It's the sort of horse called a combine harvester. Which
is of course, not a horse. It's not even an animal. It is a category
error to invoke right and left in the explanation of this phenomenon.
Fascism exists in a world parallel to that of democracy.