Most mainstream political analysis places all political positions along a single left/right political axis. On the far left, one might find Stalinist regimes, and then various liberal democratic positions, blending into a muddle of centrism. Continuing to the right, there are conservative republican ideologies and then fascism.
The authoritarianism placed at each end of this spectrum has given rise to the horseshoe theory that claims the further one goes to either end, the closer these extremes approach one another. As a result, the muddle of centrism is posited as a solution to such murderous regimes.
This strikes me as a blend of fallacious arguments, though. First, the left/right spectrum presents a false choice. Second, the fallacy of the middle ground asserts that the truth always lies between two given extremes. Third, the premise that every idea can be assigned a point on this axis is dubious.
The political compass chart makes some headway toward improving these oversimplified analyses, but it remains imperfect. It ads a second axis of authoritarianism and libertarianism, allowing a clear distinction between, say, Maoist China and pot-smoking antiwar hippies, even though both are generally considered "far-left ideologies." The authoritarian vs. libertarian divide tells me a lot more about someone's beliefs than the left/right scale. It also eliminates the horseshoe theory.
The political compass questions are flawed, too, though. there are certain presumptions that make the hard to decipher, and the lack of a "This is none of the government's business in the first place" skews the results on the authoritarianism axis considerably.
Perhaps a third axis might clarify matters. We could model the chart in three-dimensional space as a cube with left/right, authoritarian/libertarian, and some other element. But what? Theology/atheism? Emotion/reason? We could find ourselves in Briggs-Meyers levels of self-absorbed pointlessness quite quickly. At some point, the game of splitting hairs and compartmentalizing people seems like less of a clarification and more of a way to just summarily dismiss them.
Let's instead limit ourselves to the authoritarian/libertarian axis. Let the commies and the fascists bicker. We don't need to pick a side when authoritarians argue. That is a losing game every time. Do you support voluntary interactions and exchanges, even when you disagree with the means others may choose? Do you accept that others have different preferences? Do you wish to take more responsibility for your choices and their outcomes? We can talk. Maybe we can even achieve progress toward a new liberty instead of the counterfeits politicians promise every election cycle.
I've actually gone into the shortcomings of the two-axis political compass twice, listing four different positions that people can have that, despite common associations, can be held by anyone, regardless of where they fall on the political compass. Reality denial, bigotry, and violence are all vices which anyone of any political ideology can indulge in.
Personally, I think the third axis should be "status quo vs. paradigm shift," as in "how much do you prefer to adapt yourself to working within the system" vs. "how much do you prefer to change the system itself," or to put it more concisely, the axis of personal flexibility.
That could work. I want to change the system. So do progressives and white supremacists. None of us want the same kinds of change, though. And the traditionalists appeal to differing traditions.
Expanding on that idea, perhaps the centre of that axis should be "keep things exactly the way they are," while a return to tradition would be on one end (greater regression being farther toward the extreme), and moving to an entirely new paradigm would be on the other. Once again, however, we run into the exact same problem, because there are some things about "the good old days" (or "the bad old days," as I call them) that many of us yearn for, and other things that we'd rather leave in the dustbin of history. Likewise, some things that have never been tried before turn out to be great ideas, but other "progressive" ideas turn out to be terrible, and need to be reversed. There is one thing that I love telling self-described progressives: just because you're moving forward, doesn't mean you're moving in the right direction.
Is it change toward liberty or authority? Is it a tradition of independence or subservience? Honestly, I think the "left/right" axis is the least relevant compared to progression/regression and authoritarian/libertarian.