Don't forget that Mussolini started off as a socialist at first and that significants swathes of the Nazi party were very socialistic and left wing in their orientation.
The thread wasn't meant to be a debate, and it's getting quite close, so I'm going to stop after this.
Mussolini explicitly repudiated his former socialist views. To characterise fascism as left-wing because of Mussolini's background, is on a par with classing Stalinism as a form of Christian theocracy because Uncle Joe went to a seminary. And there really weren't any swathes of the Nazi party that were socialistic in outlook. Even Strasser, who is cited as being a genuine socialist for wanting the state to take control of land and industry, proposed this a stepping-stone to redistributing those holdings to an elite of good German stock so that feudalism could be restored (selective quoting in certain books leaves that bit out).
Italian Fascism did not want Italy to return to say the time of the Roman Empire, the time of the Papal State or the time of the Italian communes.
You don't think their adoption of the fasces, the symbol of law and order in the Roman Empire (the thing that gives us the word "fascism"), indicates at least some desire to return to that time?
Their outlook was clearly futurist in a lot of ways and don't forget that both fascism and nazism have always been very anti conservative and very anti bourgeois.
Their methods were futurist, but their outlook really wasn't. The Fascists in Italy and Spain, the Nazis, even the BUF, all saw themselves as leading a crusade to return to some quasi-mythic past, to a supposed golden age. I can't really see any part of the left that sees things in those terms (maybe some sections of the Green movement, if you view that as being left-wing).
Alliances between facists/nazis and conservatives were alliances of circumstances and nothing more. To a degree they shared the same political objectives in the sense that they supported nationalism, a stronger nation and so on. But while the conservatives wanted to do this through their respective countries traditions. The nazis and the fasicts wanted a clean break with the past, traditions were an hindrance for them and not something to be respected and cherished (a conservative characteristic).
Surely those shared political objectives (including opposition to democracy and civil liberty) make it clear why fascism is on the far-right of the spectrum. This is the same as socialists and communists sharing the same political objectives (at least in theory) but wanting to accomplish them in wildly different ways - communists through revolution and force, socialists through reform and democracy.
Witness for this the fact that the nazis did not reestablish the German monarchy, while most conservatives where in favour of this.
The nazis and the fascists were whether you like it or not a lot more progressive than conservatives. Witness the Kraft durch Freude organisation for workers, extensive social programs, paid holidays and so on.
Granted Hitler didn't honour his promises to restore the Kaiser. But they did restore a monarchic system of rule. And it's not to do with liking it or not - they simply weren't (politically) progressive, in any way. They weren't conservative either - the best term for them is regressive, well beyond the conservative and in opposition to the progressive. Offering inducements to the masses doesn't qualify as progressive, not least because plenty of conservatives have done it. Unless you want to exclude the likes of Bismarck, D'Israeli and Hoover from the conservative side. Did Margaret Thatcher's expansion of the right-to-buy scheme make her a left-winger?
Nazi/Fascist economics where far from laisser-faire too, they worked with captains of industry because it was a mutually convenient relationship for both and nothing more. The nazis could rearm, the industrialists made money without having to worry about competition and such. Hardly a mark of laisser faire capitalism or rather conservative economics.
Laissez-faire was liberal economics, and (European) conservatives at that point still advocated protectionism (some were even still advocating mercantilism, perfectly in-keeping with Nazi economic programmes).
Lastly the role of the church was never emphasised in Nazi/Fascist propaganda. Some nazis were very much Pagan and most did not give a shit to Christian morals or principles. Reactionary regimes like Franco's Spain and such such where reactionary but not fascist. Social welfare programmes were limited or non existent, links with the past where constantly used to bolster the regime and institutions like the church played a key role in the running of the state.
The role of the Church may not have been emphasised (though I think sometimes it was by every fascist strain except Nazism, and the Nazis were presiding over a country with two significant denominations), but the Nazis and Fascists frequently claimed there was a need to defend Christianity from Bolshevism, Liberalism and Modernism (and that only they could do this) - and this was one of the main planks put forward by their apologists in the West. The Falange was very much a fascist organisation. It had unique qualities, but political movements manifest themselves very differently in different nations (compare the Conservatives in Britain with the Republicans in America). Where it differed most from other fascist movements, was not in its relationship with the Church but in its partial adoption of syndicalism from the idiosyncratic Spanish left. And of course, Franco didn't restore the monarchy until much later.