There isn't really an "American version." Left vs right is broadly okay if framed as collectivized ownership as principle vs privatized ownership as principle, but economies in the real world aren't "pure," and trying to gauge how left or right a country is by proportion of the economy that is public vs private can be misleading. The next part, "libertarian vs authoritarian," is a false binary. The state is thoroughly linked to the mode of production, you don't just pick something on a board and create it in real life. There's no such thing as "libertarian capitalism," as an example. Centralization vs decentralization may make more sense, but that can also be misleading, as centralized systems can be more democratic than decentralized systems.

This is a pretty good, if long, video on the subject. The creator of the compass is also politically biased.

As a fun little side-note, I can answer the standard political compass quiz and get right around the bottom-left while being a Marxist-Leninist that approves of full collevtivization of production and central planning. Yet, at the same time, the quiz will put socialist states in the top left, seemingly based on how the creator wants to represent things. It's deeply flawed. Add on the fact that it's more of an idealist interpretation of political economy than a materialist one, and you've got a recipe for disaster.