The Political Compass is a popular "test" on the internet that supposedly reveals your true political "position". Instead of the traditional "left-right" scale, it has a two-dimensional grid that looks like this:

Unfortunately, hardly anyone points out that this improved model is also extremely limited. A third dimension, like "Militaristic/Pacifist" could be added. A fourth dimension, like "Nationalist/Globalist" could be added. You could even add "Free-Lover/Prude". Many more dimensions could be added, and, with each dimension, you could add a "scale" to indicate how important each issue is to you. You could also add non-linear scales -- for example, you can tolerate the neighborhood watch and the police, but you draw the line at the National Guard patrolling your city.

In short, you could measure every single possible point of disagreement, and the degree of disagreement.

The resulting charts could not even be drawn on the screen, because they'd have so many dimensions!

Motivation

What then, is the reason for this second dimension?

This "test" is not an objective test, but has a political goal: to create an alliance between economic conservatives. In recent times, the "left" in the USA has been a mix of social democrats (reformist socialists) and cultural liberals who were fiscally capitalist. Generally, because the USA is the largest capitalist country in the world, capitalist economics is taken as the norm; also, because there is (or was) a large middle class, there is not a strong class division, and people tend to divide themselves culturally.

The goal of the chart is to re-divide people by their economic beliefs. Specifically, it's to leverage the 50+ years of anti-communist Cold War political rhetoric that has led to almost every American declaring themselves some kind of "anti-communist." Americans may not have felt like they are "capitalists", but they were opposed to Stalin. (Indeed, the Left in the US is mostly anti-Stalin.) They were anti-communists.

The goal of doing this is to increase the political power of capitalists. This is the effect of a group of people gaining some kind of collective self-awareness, as ideological "capitalists," and associating themselves with that group's poltical platform.

The chart is not neutral -- in America, it is a political organizing tool for idological capitalists.

However

My assertion applies mainly to the situation in the USA. In other situations, it could be a way to organize ideological anarchists. For example, if both the socialists and capitalists are authoritarian, it could serve to unite anti-authoritarians across two divided parties.

My overarching point is that this chart is not "accurate" -- it's just more accurate than a one-dimensional chart. It's not objective -- it's just a tad more objective. The bias it introduces is intentional. It's unfortunate that so many people think the chart is "objective" and aren't reflexively questioning the goals of the chart (and the sites that promote it).