The terms "left and right wing" actually originated during the French Revolution. The various legislative bodies were initially seated from right to left based on their social rank, and that slowly evolved into our modern political spectrum. So if anything, the factions of the French Revolution are easier than modern groups to pin down on the left/right scale because that's exactly how they identified themselves.

TL/DR. If we look at leftists as being for greater equality and rightists as being in favor of inherited/traditional privilege than we can (sorta) use these terms pretty broadly. However, we should probably find better, more specific terms to use.

I'm sorry if this doesn't answer your questions, but left vs. right is a very slippery concept, especially in modern politics.

As others have said the concept actually dates from the french revolutionary period. At that time people who were for greater equality and rationalism were called "leftist" and people who for inherited privilege, the aristocracy, tradition and social stratification were right-wing.

While modern leftists are still are generally pro-equality and modern rightists still generally defend inherited privilege and tradition, things have gotten all messed up since then, both because the ideals of the jacobin leftists won (no significant group of people, even on the far right, wants to bring back the absolute monarchy) and because so much has changed.

Take laissez-faire capitalism. Back in the 18th century Monarchs and aristocrats intervened in markets in a regressive way to keep peasants on the land and business people poor. Aristocrats were barely taxed at all, and the King lived off tax revenue from the poorest people. In this situation laissez faire capitalism challenged the powerful and gave hope to the powerless.

These days business people have roughly the same position as the aristocracy did, and socialist ideas have shown that it is (at least theoretically) possible to use market intervention to lower inequality rather than exacerbate it. Since laissez faire capitalism will (depending on who you ask) help rich people stay entrenched and leave the working class unprotected, leading to less equality and greater social stratification it is now considered right wing.

As far as your question about anarchists and fascists, this really underlines how useless terms like left and right wing are. Some anarchists consider themselves right wing, some consider themselves left wing and many opt out of those terms all together.

Likewise, fascism isn't particularly right-wing (they believe in a massive reorganization of the social order, are generally populist and generally want to do away with traditional institutions) but they aren't really left-wing either (they believe in social stratification, racism and are generally anti-rational).

EDIT I should mention that since the rise of classical/neo-liberal economics, and the rise of the capitalist/business class at the expense of the aristocracy, many right-wingers would say that they aren't pro-tradition/anti-equality, but rather pro-efficiency/anti-equality. Obviously this muddles the waters even further.

As a follow up, what were the political 'parties' in Rome, Greece, etc. , and how did they work. I seem to recall that Roman culture had massive organization around chariot race clubs that extended into business and politics.