Rotate the compass 45° clockwise, name the new vertical axis to state economy vs private economy, and name the new horizontal axis to left and right. At the far left(green) you have total destruction of hierarchy and flattening of power inequality, while at the far right(blue) you have absolute hierarchy and centralized power.

State centralized economies(red) lead to increased hierarchy by empowering government officials to become powerful lords. Privatized economies(purple/yellow) lead to increased hierarchy by empowering capitalists to become powerful lords. Trying to reach either the top or bottom tip of the diamond leads you the right tip.

Horseshoe theory kind of applies to Marxist-Leninists like its pro-capitalists creators imagined, but they didn’t realize that they were the other end of the horseshoe. This is because both capitalism and the state are the dangers. The problem with both stems from allowing the unrestricted accumulation of power.

Government officials use political capital, while capitalists use economic capital. They can feed off of each other, with the rich helping politicians and the politicians helping the rich. My preferred solution to both is to redistribute capital from the rich directly to the poor, growing the middle class by putting a ceiling and floor on the rich and the poor.

It’s fundamentally wrong, and it is wrong on purpose.

Leftism IS egalitarian. It is fundamentally anti-authoritarian. It is concerned with ensuring that power (and wealth, which is inextricably linked with power) are equally distributed. Anarchism and Direct Democracy are forms of “extreme” leftism.

Rightism IS authoritarian. It is fundamentally anti-egalitarian. It is concerned with consolidation of power and wealth. Monarchism and Dictatorship are forms of “extreme” rightism.

Economic policies are completely beside the point.

A dictator who nationalizes things is still right wing, and has merely brought those things under his own personal control. “Benevolent” Rightism is still rightism.

A leftist society that votes to nationalize things is still leftist because they have achieved a quorum and share the control equally.

Rightism needs to muddy language and understanding to achieve enough support to seize power. “I am a leftist” some will say as they seize power for themselves and entrench themselves like a monarch or ruling class. They are rightists and they know what they are doing.

“Look at those authoritarian leftists” others may say, as they point to extreme-right power structures who may or may not call themselves “leftist.”

Social and economic policies are often thought about as left/right on their own, as some may be seen as advancing either egalitarian goals or authoritarian goals, but left-seeming policies under a rightist regime is merely self-serving for those in power and easily changed without the consent of the populace… and therefore not leftist at all. Likewise, even a leftist society could enact policies that could be anti-egalitarian, though such policies would put that society at ever-increasing risk of drifting right as power is gradually stolen from the people.

TL;DR;

The political spectrum is entirely about power and who wields it. Specific policies are utterly tangential since they are nothing but tools wielded by who is in power… an entrenched, consolidated entity… or the people themselves. The political compass twists this into knots in attempt to bolster “both-sidesism” and distract anyone exposed from the true and simple nature of rightism: hoarding power. When the people understand and can talk clearly and unambiguously about these topics, the entrenched powers stand to lose their privileged, predatory advantages… and they will fight against that with everything they have.