The general problem is that the U.S. seems to have adopted a political jargon that shares a lot of the terminology used in political science around the world, but applies some "american exceptionalism" towards it's meaning so that in the end they sometimes fit, sometimes don't, but usually serve to confuse the heck out of anybody who is accustomed to how words work. And I don't just mean elsewhere, it also seems to confuse Americans themselves too.
But enough of a rant, let's get to what these words actually mean (afaik; and to countries other than the U.S. as no such tag is applied to this question).
Now the origin of the left-right distinction is usually the National Assembly of France following the revolution. In it people with similar opinions took places next to each other (presumably to coordinate their proposals better) and so "left wing" and "right wing" literally referred to people on the left or right side of that room.
From there you had abstractions and categorizations. So people gathered the positions that were common on the left and the right side and constructed various spectra where mutually exclusive positions were on the borders and where political parties, ideologies and actors were placed on that spectrum. Idk republican vs monarchist, freedom vs authority, egalitarian vs elitist, international vs national, cooperative vs competitive and so on.
So one that is as far as I know used in social science today is one of social hierarchies, so something of a variant of egalitarian vs elitist. Where right wingers believe in some sort of natural hierarchy where people have their place and are supposed to stay there so that things are nice and ordered, while left wingers are in favor of egalitarian structures where people have equal agency and where the vivid discourse is not seen as chaos but rather as freedom.
So if you are familiar with that god awful political compass, this would be roughly the diagonal from bottom left to top right, while the top left and bottom right sector would make no sense and you'd rather take the projection to the diagonal by the coordinate that is most extreme. So if they are peak authoritarian they are top right no matter where they are placed on that poorly defined economic axis and if they are far right on that economic axis they are also top right. Because all that talk about freedom is moot if you are in favor of a Social Darwininism that orders society along a hierarchy, where some rule the others. Doesn't matter whether that's based on money, race, sex, ethnicity, religion, ... or whatever other excuse you feature in order to mistreat your fellow people.
Now Anarchism is the believe that people can get away fairly well without leaders and ought to rather organize themselves. So they are really anti-authority. No god, no state, no slave, no master might be a popular shorthand.
So if you don't want to rule and not being ruled the result is logically something egalitarian without social hierarchies. Hence Anarchism (ideologically) is always bottom left. Now I write ideologically, because obviously just because people see themselves as proponents of something doesn't mean they actually end up like that. Like lots of "liberals" (in favor of individual liberty) also were slave owners and colonialists, so much for the liberty of the individual under their boots...
But at least in theory Anarchism is supposed to be a the most egalitarian as anything else would mean that someone is ruling the rest. Now there are different schools of anarchism that vary on their focus on the individual or the organization of society. So is society a cooperative community or rather a "union of egoists" that has no purpose in itself, but is mutually beneficial to it's members.
Though both of these classical anarchists would still be on the furthest left. And economically Anarchism would be socialist.
The general idea being that economic disparity is the OG of political power control and coercion so even the right and centrist anarchist came up with slogans like "what is property? PROPERTY IS THEFT!". And so people should either be able to live self-sufficient or if they form collectives should be equal share and stakeholders.
So from the conception of classical anarchism, the U.S. libertarianism or Anarcho-Capitalism would not be a school of Anarchism. And if you read their Wikiarticles that disagreement is mutual. Neither claims the other is in any way shape or form related. (Same for other combinations of anarchism of right wing ideologies, at least to my knowledge there is not one with a consistent definition that isn't doing one part of it's description massively wrong and mostly it's the anarchist part.)
They operate on fundamentally different principles and just because the Libertarian/AnCap faction stole the terminology does not mean that they refer to the same things when using them. And yeah they stole these terms, they were centuries late to the game (originated in the 60s/70s), had their origins in Old-Right conservativism and even admitted their pleasure of co-opting left wing terminology.
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...”
― Murray N. Rothbard, The Betrayal Of The American Right
So yeah if you refer to them then that would be a right wing ideology. Though again not because of their stance on "big government" but rather because they praise property to an extend where it allows enslavement, exploitation and basically results in tyranny and a neo-feudalism. Like seriously they place no restrictions on the haves and provide no protection for the have nots and no reasonable means for to bridge that gap. And no that individual transition between classes is not impossible does not mean that you're not living in a caste system. That is as far away from the idea of no one should be coerced or coerce others as can be. Their anarchy refers to non-regulation of the market not to the freedom of the individual.
Also what does "small government" actually mean. Like a state that is reduced to military and police to uphold a status quo that is fundamentally unequal and unfree for a majority of people is NOT the least amount of government interference.
Again they only focus on the market not on the people, which makes these claims nothing but absurd.
And apparently to avoid the fitting label of right wing and to unjustly dress themselves in nicer language they basically turn the political spectrum by 180 and pretend the left right spectrum goes, with respect to the political compass from the top left to the bottom right. So from most state controlled economy to most market controlled economy. That the economy is controlled nonetheless and that the individuals are still by and large unfree is completely ignored by that picture.
Further using left and right to refer to the overton window rather than an abstraction which actually spans a full spectra isn't helping in analysis either, but that's unfortunately not a U.S. specific problem but something done by many countries on a regular basis.
That should give an overview over what these terms mean an how to apply them in your particular example whatever your definition of the terms should be.