I once heard Noam Chomsky describe himself as one. I have never heard the term before. If it is a valid term, what does it mean beyond the obvious mash up?
Yes, anarcho-socialist is a valid term in political science.It refers to communities that organize along communitarian grounds without an overarching set of governmental institutions, and is usually opposed to anarcho-individualist or anarcho-liberalist theories, which base themselves on mutual interactions of individuals. One might think here of the ideal form of communism (not the state-authoritarian forms of communism we generally find in practice).
Philosophical anarchism generally aims at some sort of trans-government mutuality — an abstract form of social contract theory — in which people voluntarily commit to maintain certain rules of social order without the need for force or punishment. The various flavors of anarchism come from the ways in which that 'mutuality' is conceptualized: whether it arises through market forces, moral obligation, social indebtedness, common effort and labor, etc... It is idealistic and utopian as a rule.
Author's note: if you comment, please try to avoid discussing your (or anybody else's) personal value judgements on ideas about who should have power, how production surplus and rent should be distributed, and the like. Such discussion adds more heat than light when one is merely trying to describe the philosophies between the systems in question. I certainly have my biases, and some may come out in this answer, but I don't ask that you agree with any you should happen to see; if you do disasgree, take it as granted that I accept your value judgements as valid and that you think my value judgements are wrong.
Yes, "anarcho-socialist" is not only a valid term, but a necessary one to describe what sort of anarchist someone is amongst the extremely broad range of forms of anarchism.
Anarchism is described by Wikipedia as, "a political philosophy and movement that is skeptical of all justifications for authority and seeks to abolish the institutions they claim maintain unnecessary coercion and hierarchy." Chomsky describes it in similar terms in this group discussion at McMaster university (0:21):
Anarchism covers lots of different things. If there's one leading principle which unifies them it's a simple one: it's based on the assumption that any...structure of authority and domination...has to justify itself: none of them are self-justifying.
Note carefully that this does not mean that anarchists never accept authority and even coercion: it means only that the anarchists together must agree that delegating power to that authority is a useful compromise to achieve their aims.
The level of authority and coercion considered "justified" will often be highly dependent on other, orthogonal political views, as well as the particular situations that an anarchist group finds itself in from time to time.
As just one example, should an anarchist society be faced by an external threat that could destroy their society, they might agree that in order to fight this threat in what they believe is the most efficient way they should set up a conventional, very authoritarian military with significant power over almost all of society, along the lines of what the western nations did in World War II. If doing this is mutually agreed upon by the members of that society, and later change in this form of government is not precluded, this is continues to be an anarchy, despite having on the surface having an authoritarian form at that particular moment. The key here is that the members of the system voluntarily set up an authority and gave it specific power, and still have the ability to remove that authority when they feel it's right to do so.
It may help to think of anarchism vs. archism as a third spectrum, or at least flavour, to be added to the classic left vs. right and libertarian vs. authoritarian grid. Whereas the libertarian/authoritarian spectrum is concerned with power ideology, such as how much concentration of power there should be and where it should lie, the anarchism/archism spectrum is concerned with the source of the power, rather than the concentration of it. This is indeed a quite subtle distinction, but the examples below I hope will illuminate it.
Anarchism is often thought to be a form of socialism that falls within the left and libertarian corner of the grid, and this is indeed where the "anarchosocialist" philosophy lies. A primary point of socialist economic philosophy is social ownership of some part of the group's wealth. This might be, for example, that the surplus generated from land (extracted as economic rent) should in whole or in part be returned to the community in some form. (This can be done via a wide range of means, from taxation to full-on public ownership of land; even "socialism" itself is a spectrum, not a point.)
But we could also combine anarchism with a more right-wing libertarian philosophy to get anarcho-capitalism, which, "seeks to abolish centralized states in favor of stateless societies with systems of private property enforced by private agencies, the non-aggression principle, free markets and the right-libertarian interpretation of self-ownership, which extends the concept to include control of private property as part of the self." (Chomsky describes the vast difference between this and his preferred societal structure in Chomsky refutes "libertarian" "anarcho"- capitalism; he also illuminates some of the issues with the term "libertarian," which has had rather different meanings in the U.S. now vs. elsewhere at other times, asking and answering the question "How can I call myself a libertarian socialist?")
As well as the huge difference between these on the left/right spectrum, you might feel you also notice some difference between the two on the libertarian/authoritarian spectrum: anarcho-socialism seems to limit theoretical personal (economic) freedom¹ much more than anarcho-capitalism does, and thus seems more authoritarian.
This is where the distinction I mentioned above between anarchy/archy and libertarian/authoritarian appears: anarchy is concerned not so much with the actual amount of personal freedom you have in its various dimensions (though it usually defaults towards increasing it without countervailing reason) but with your agreement to the levels of various kinds of personal freedom versus other benefits you can get from society.
For comparison, it's also possible to have a very right-wing, economically libertarian society that still has an authoritarian core that can not be changed: there might be a ruler that dictates economic libertarianism who cannot be changed or removed. Less anarchic libertarians might be ok with this because they're concerned only that they have the particular freedoms they value, and they're not too concerned about from whose authority (their own or others') those freedoms come.
¹ I say "theoretical" because obviously a poor person within such a system will have much less freedom in practice than a rich person will. How important you think this is is a value judgement, but surely we can all agree that "I can keep all wealth I gain through whatever means and spend it how I want" is indeed greater personal economic freedom for that person than "some of the wealth I gained has to go back to the society within which I gained it," regardless of whether we think of that as a good or a bad thing.
It's a rather uncommon term, like Wikipedia would only use it as a synonym for libertarian socialism. But if you mean whether it would make sense, then yes.
Socialism is a broad umbrella term which includes: "anarchism, communism, the labour movement, Marxism, social democracy, and syndicalism".
One objection might be that in the 19th century anarchism and socialism were used interchangeably so that it would be unnecessarily double the words.
Recently (1960s) apparently U.S. conservatives tried to take over the term Libertarian and establish one called "Anarcho-Capitalism". As this quote from Murray Rothbard (apparently famous "libertarian" figure and found of anarcho-capitalism) exemplifies:
"One gratifying aspect of our rise to some prominence is that, for the first time in my memory, we, 'our side,' had captured a crucial word from the enemy. 'Libertarians' had long been simply a polite word for left-wing anarchists, that is for anti-private property anarchists, either of the communist or syndicalist variety. But now we had taken it over."
These attempts are sometimes falsely attributed to individual anarchism but either side rejects the other.
Anarcho-capitalism is also distinguished from anarchism, an anti-capitalist movement that opposes unnecessary coercion and hierarchy, and social anarchism, a branch of anarchism that sees individual freedom as interrelated with mutual aid. Unlike anarchists, anarcho-capitalists support private property and private institutions
Traditional anarchist schools of thought oppose and reject capitalism, and consider "anarcho-capitalism" to be a contradiction in terms, although some, including anarcho-capitalists and right-libertarians, have argued that anarcho-capitalism is a form of anarchism. Anarcho-capitalism is usually seen as part of the New Right
In both its social and individualist forms, anarchism is usually considered an anti-capitalist and radical left-wing or far-left movement that promotes libertarian socialist economic theories such as collectivism, communism, individualism, mutualism and syndicalism. Because anarchism is usually described alongside libertarian Marxism as the libertarian wing of the socialist movement and as having a historical association with anti-capitalism and socialism, anarchists believe that capitalism is incompatible with social and economic equality and therefore do not recognize anarcho-capitalism as an anarchist school of thought.
Anarcho-capitalists are distinguished from the dominant anarchist tradition by their relation to property and capital. While both anarchism and anarcho-capitalism share general antipathy towards power by government authority, the latter exempts power wielded through free-market capitalism.
Rothbard argued that anarcho-capitalism is the only true form of anarchism—the only form of anarchism that could possibly exist in reality as he maintained that any other form presupposes authoritarian enforcement of a political ideology such as "redistribution of private property", which he attributed to anarchism.
And those are only from the anarcho-capitalist article. So it's safe to say that despite the similarity in name Anarchism and anarcho-Capitalism have pretty little in common and are generally using different definitions of anarchism and have vastly different goals and that the socialist branch of anarchism is the original one.
And although there is a soft and/or hard split in anarchism between individualists and collectivists they neither claims Ancaps on their side and again Ancaps reject both of those. So to group them in the same category would be a disservice to both.