Are the Nazi Party, Communist Party, and Bolsheviks far-right or far-left?

You are asking as if those terms had a clear scientific or at least philosophical meaning. They are ideological, polemical, and sometimes superficial terms.

What left and right means in politics is complicated and it may vary depending on the country and context.

Already the Wikipedia article mentions this complexity.

Being vague doesn't stop them being used anyway in real life, mostly in journalistic, electoral, or other contradictory arguments, but cannot be used in a serious philosophical argument, and no scientific significance can be ascribed to them. They have been created for the purpose of political debate, a purpose that is in many cases propagandistic and polemic, not descriptive or explanatory.

Their meaning is created on a "spectrum" based on opposition between parties in relation to fundamental values, ideals and institutions. These values, ideals, institutions that represent the "apple of discord" may vary depending on the historical context, and thus the meaning of these terms also varies.

There are many partially different "spectrums" where these terms take somewhat different meaning according to the object of discord:

• Monarchy: the original French meaning, in relation to the faith of French monarchy: left being anti-monarchical or republican, right being pro-monarchical; that meaning is still partially present in French politics but was largely replaced by the others;

• Socialism: the meaning in relation to the socialist movement and the "class-struggle" concept, the history of capitalism and socialism and the recent pro-capitalist reforms in the ex-communist states; this is the meaning that is mostly present in Europe, where "right" means "pro-capitalist" or in favor of free enterprise, and "left" means in favor of economically disadvantaged people;

• Nationalism: where patriotism and national identity is exalted by the "right" in opposition to its adversaries ("left") for whom this is not the main concern.

• Democracy: when democracy is at stake or under attack, its adversaries will impose the meaning of "right" and "left": the far-right will consider the democrats being "left", the far-left will consider them as "right".

It is obvious that other topics may create different "spectrums", some of them outdated (colonialism) and some new (ecology), but what is certain is that "right" and "left" have different meaning in the different "spectrums", the same party or opinion may have different qualities (right or left) depending on the spectrum, but also they may have the same quality in different "spectrums" without these qualities having the same meaning.

The main complication comes from the fact that these "spectrums" are superposed for practical or ideological reasons but in many cases without the full awareness of those involved in the debate; thus the meaning of our terms become confusing, even contradictory.

Things get more complicated by the fact that in time certain elements or institutions that were present (and determined the meaning of these terms) fade away or, on the contrary, they stop being subject of debate and are taken for granted. For example, monarchy is no more a subject of debate in France. Democracy is not a subject of debate in many democracies, like the United States, Australia, New Zealand or in parts of Europe (where no party promotes dictatorship); in such countries "right" and "left" take their meaning from other "apples of discord".

In some cases other basic terms have different meaning based on the context; in the US "liberal" and "left" mean almost the same thing, close to "socialist", while in Europe "liberal" is used in relation to political and economic Liberalism.

In order to directly answer the question: the Nazi party and the Italian Fascist party can be considered far-right, the Russian Bolshevik and all real-Communist parties ("real" as in Realpolitik) can be considered far-left in most of the theoretical "spectrums" described above.

If you are also talking about the more or less democratic communist parties or factions (in the sense of the "democracy as we know it", not in that of the "popular" democracies; I will not touch the problem of how "democracy" can mean "dictatorship" in real-Communism) then there's a different matter, as those cannot be then seen as "far-left" or "extreme", but come closer to the West-European socialism and Social-Democracy. (Your question is vague, simplistic, and about to be closed for good reasons.)

You seem to wonder whether the two categories of parties could be placed within the same area of any of the "political spectrums". Could they? Surely not in the cases I mentioned.

The only case imaginable would be one in which they had common goals and a common adversary, and that would be the rejection of democracy "as we know it" (liberal, "bourgeois") in favor of an authoritarian project. But there, as I said, as they impose the extreme meaning of "right" and "left", these "extreme" parties are at opposite ends of the spectrum (with the democrats at the center). While socialism vs capitalism, republic vs monarchy, and internationalism vs nationalism can be described as a left-right definition of various types of a political spectrums, democracy vs authoritarianism has not produced in reality a such spectrum in which fascist/nazis and bolshevik/communist would be at the same extreme of the spectrum. Even then, each of these extremist parties would give opposite descriptions of their own position. A democrat could be imagined putting them side by side at one extreme, but it is difficult to say if that far-region of the spectrum should then be called right or left, or whether these terms keep their meaning in a such case.

As right and left serve mostly propagandistic and polemic purposes in real political adversity, the parties that are most aggressive ideologically tend to impose the most pervasive meaning of these terms, given that the ideologies of such parties need clear and categorical, even simplistic and one-sided terminology. But in order for a such discussion to even have a meaning such parties have to be present within the same spectrum. They can do that only theoretically, and, as I have said, these terms have a poor theoretical value; in reality both the Nazi and the Bolshevik (and other Communist parties of the "real socialism") had as main concern the fast resolution of the debate through the physical elimination of their political adversaries - and of any political diversity in fact. After that, the use of terms like right and left becomes useless or changes completely its meaning, like in the case of the persecution, prosecution and extermination by Stalin of his (real) "left" and (imaginary) "right" opposition (the "danger from the right") within the Communist party.