I read some exotic political theories, around here. Well, I guess we need some kind of glossary to debate.
Indeed the usual left/right classification was born during the french revolution : The words themselves refer to the side of the Convention nationale where our first representatives seated. However, we had much more exotic categories at that time, with jacobins, montagnards, girondins ... The nowadays classification is a result of Louis XVI's execution vote : The left side voted yes, the right side no. So, until at least WWI, this division means progressive republicanism (left) against social reaction(right), be it republican (most for the sake of compromise than real fervor before the early XXth century) or monarchist. For instance, the Parti Radical was considered as a leftist party, while it defended a brand of demoliberalism. The original left is liberal, a reaction to royal and aristocratic abuses, most notably against property, which angered the bourgeois class which seen nobles as economic parasites. Excepted Germany (Well, Bismarck would disagree), socialist parties were seen as political weirdos and threats by both left and right.
Ironically, socialism owns much to Bismarck : While he intented to bar it from power by "stealing" and applying its agenda (Workers regulation, unionism ...), it proved the efficiency of such reforms and the benefits a country could extract from the newly found prosperity of its subjects (Death and Taxes).
To resume socialism (by the terms of its inventors, not post-liberal revolution description), it aims to establish an egalitarian society (the terms are important, egalitarianism is not the same as supporting equality) by the seizure of the means of production, and in truth the means of power. This seizure, it's communism, when workers put every part of economy, politics, and military in common for the betterment of society. So : Base society --> Communism --> Socialism. Because the goal of socialism is to unify all proleterian classes all over the world, the concepts of nationality, religion and cultural antagonism are rejected, as well the establishment of a social, economic or political hierarchy. Ideally, socialism means an unified humanity ruled by councils of equals, without any part of society less considered than another.
The core belief of nationalism, on the other hand, is quite similar originally to socialism, which explain the confusion of the two (Which aren't exclusive) : From the 9th century to the 19th, Europe was divided between rival families, and common identity had virtually no weight, religion excepted (which did matter, but was also ruled by dynastic interests). Nationalism permitted low born population to find a common interest in power : The land belonged no more to a person, but to the group. When the First French Republic discovered a French Nation, it was in fact a community. Nationalism became the favorite meal of authoritarian regimes because it promised progress of the community through the wise guiding of chief, in the same way capitalism became the meritocratic excuse if the powerful (we rule because we are the strongest).
Fascism can be leftist, but national socialism cannot. The first is an unwanted child, the second a fraud.
Fascism is the believe that perfect equality can only be achieved with the guiding hand of the state and, to guarantee the operation's sucess, need to bind every aspect of society to its will. Mussolini was indeed a communist, originally, and he was known before WWII to laugh at idea of cultural superiority or racialism. He wanted to reestablish a Roman Empire, and the Roman Empire was fond of multiculturalism and syncretism (Just consider the slow evolution from "greek" republican polis system to a persian divine imperial regime, or the popularity of the cult of Isis). Yet, in the same way Lenin used the tsarist reference to assert his personnal power, Mussolini used racism to explain his regime's own failures.
National Socialism has an original flaw, disqualifying it from being branded as leftist : Its core belief is the superiority of a minority over the rest of humanity (Aryan supremacy, völkich influences, establishment of a strict social and racial hierarchy), placing it in total opposition with socialism (Remember the Internationale motto : Workers of the world, unite !). While Stalin supported "communism in one state" to evade Trotsky's internationalist shadow, Hitler had always been in favor of supremacist Grossdeutschland, whose conquests were to serve the survival of german population, and germans only. Some of you will point out the name of the NSDAP. Just remember the context : Weimar Republic was still recovering from the Spartakist uprisings, Kapp coup shown that nationalist monarchism was still powerful in Germany, and the society was deeply divided and its national glory tarnished by the war. NSDAP played on both inequalities issues and national proudness. After its electoral success, there was no more socialism or illusion of socialism in its practice of power, however : THe economy was at the hands of powerful industrial ententes (The Krupps are the best exemples), much like the japanese Zaibatsu, and access to the administration was conditionned by racial and political criterias.
To rationalize a bit this agressive block of text : Fascism may be an hybrid of leftist agendas and nationalist reference, yet it constitutes an original ideology, the term "Third Way" is accurate to describe it. National Socialism, however, is only a mere reactualisation of the XIXth century antisemitic monarchism (The führer is a monarch, yet not a royal) sustained by the original pre-WWII german corporatism.
Edit : "Implemented policies. Self-declared socialist could be considered a right-wing if he implements right-wing policy. Because actions are more important than a self-designation".
This pal above me said in one line what I wanted to explain with a whole book. Damn you.