What is 'Anarchy'?
Anarchy: Originally a term used to describe leaderlessness, anarchy later became a political movement that rejected the concept of government, instead advocating for stateless societies based on voluntary association. Opposition to the concept of the state is core to anarchism, but many anarchists reject any form of human hierarchy, state or otherwise. It's at this point that anarchy reaches the point of becoming a logical paradox. Without any form of hierarchy, there can be no claim of value and therefore the anarchist's value claim that hierarchy is 'bad' ceases to have logical weight.
In this original political form, anarchists varied between those who retained the presupposition of egalitarian universalism; that all human life shares equal intrinsic value, and those who adopted a more Darwinian view. Abraham Lincoln, himself a libertarian leaning liberal, described the latter form of anarchy as 'licentiousness', and defined it as 'those who exercise their own liberties without due regard for the rights and liberties of others'. In this 'extreme individualism' form, anarchy forms the extreme 'right wing', in the form of an 'extremist individualist'. Despite this important differentiation between forms of anarchy, the label was usually subsumed into the larger 'libertarian' label during the 19th century and first half of the 20th century.
When democratic socialists and fascists began to rebrand themselves as 'modern' liberals in the mid-20th century, liberals, particularly in the US, began to rebrand themselves as 'libertarians' in order to differentiate their liberal philosophy from the increasingly anti-liberal tendencies of 'modern' liberals. Once again, 'anarchists' began to be differentiated from 'libertarians', with the former fitting Lincoln's description of 'licentiousness' and the latter simply being classical 'pure' liberals.
However, another definition of anarchy has been adopted by those who adopt a view derived from philosophical naturalism. In this version of anarchy, egalitarian universalism is rejected in favour of the Marxist position, that human life has no intrinsic value. The result is a far darker and nihilistic philosophy. Originally a fringe belief, this form of anarchy gained wider acceptance during the 1960s and 1970s among former Marxists like John Paul Sartre, the French existentialist and playwright. The result has become 'anarcho-communism' and 'anarcho-syndicalism' among other variations of 'socialist' anarchism.
In this latter form, the term 'anarchy' has increasingly been seen as a 'left-wing' phenomenon. Once again we face a logical paradox, where anarchy (extreme individualism) is combined with socialism (extreme collectivism).
Long story short, anarchists are extreme individualists who reject the authority of government or other hierarchies. This places them at the extreme 'right wing' of the political spectrum.
'Socialist' anarchy is a term adopted by those who were forced to distance themselves from the genocidal record of both egalitarian socialism (communism) and anti-egalitarian socialism (syndicalist socialism or fascism). In doing so, former Marxists could pretend to reject the hierarchies 'responsible' for genocides that cost 180-250 million lives between 1900 and 1970, while firmly holding on to the philosophy that created them in the first place. Textbook cognitive dissonance, not anarchy at all but neo-Marxism in thin disguise.
Reading List -
Abraham Lincoln and the Second American Revolution, McPherson, James M.
Edited: 26 Dec 18