Here I made a version of political compass with annotations for each quadrant. There are nine quadrants in total as we need to accommodate various centrist tendencies, such as lib center as well as dead centrism.
There are three columns and three rows, combination of column and row produces the entrain feature of said ideology.
Collums are Arbitrary Rule, Rule or Law and Ochlocracy
Arbitrary Rule simply means that person one or another way in charge can make whatever decisions they please.
Rule of Law means there is law that either guarantees certain rights and freedoms or in contrast limits said rights. Some form of rule book does exist that all must abide by.
Ochlocracy is unrestricted rule of the majority where majority may vote to exterminate minority is they so wish.
Rows are Illiberal (Authoritarian), Liberal and Libertarian
Statist (Illiberal or Authoritarian) means that authority does not recognizes any limitations on their power.
Liberal means they recognize basic human rights and freedom.
Libertarian (Autonomous) here simply means unrestricted. That word means different thing based on quadrant.
Now on how each combination of these principles turns out to be.
Authoritarian Arbitrary Rule (top right) is basically one man rules the state as he sees fit. It could be either absolute monarchy or a dictatorship. It could be a group of people on top, such as Military Junta. Ideologically its best represented in ideas of Charles I or Luis XIV where King is the state and can do no wrong as state is his private property. I do not think anyone expresses such ideas about governance openly nowadays.
However, this system can sometimes exist as a military dictatorship where support of the military ensures the rule of the strongman. It can stay stable while leader has loyalty of the military and opponents are too weak.
This is tyrannical and unfree and does not even try to hide it.
Authoritarian Ochlocracy (top left) Ochlocracy is rule of the mob. Its difference with democracy is not in principle but in outcome. Unrestricted tyranny of majority (dictatorship of the proletariat in Lenin's terms) will always lead towards tyrannical outcomes. It's a system where two wolfs and a sheep vote on what to have for dinner.
Eventually it always devolves into a illiberal charismatic dictatorship where a leader, propelled to power by public support would use this power and public support to silence all opponents to shore up his position at the top. Due to lack of security in his tenure and fear of losing power the leader will silence any and all potential opponents to his rule to avoid them gaining enough popularity to take his down and take his place.
It's a system where only those who can believable accuse everyone else of being a Nazi and then send them to the firing squad before they could do the same to them. That makes it even more oppressive than Autocratic Arbitrary Rule where leader, more confident in security of his tenure, can afford to take it easy.
Stalinist USSR is example of this system.
Illiberal Bureaucratic Despotism (top center) Rule of Law tend to be associated with Liberal Democracy, but it can be authoritarian if laws are illiberal. Stable bureaucracies tend to drift towards this system. The bigger and more layered the bureaucracy becomes and the longer it exists unchanged, the more bureaucrats can find out all the ropes and figure out how to rig them. With that they can erode checks and balances of democracy and essentially become an unchecked autocratic system. Talk about politicians being out of touch with people are signs that system has drifted into bureaucratic despotism.
This system often looks like a paternalism where state treats its citizens as clueless children and sees themselves as benign parents. It maintains plethora of laws, rules and regulations that bureaucrats think is for the best of society. The laws however serve bureaucrats a lot more than the rest of the society. Many of these laws are aimed on preserving status quo of the bureaucracy that enacted them.
Overall, this regime is characterized by the oblivious out of touch government that has little idea how actual people live but nonetheless believe they know better what is best for them. It's as sclerotic as long past retirement old person. More often than not it stuck in the past and unable to effectively respond to contemporary challenges and issues. If left unchallenged it will devolve into caste system, where different classes of people lead vastly different lives. Leadership caste will prosper, often oblivious to life outside of their bubble while lower castes would literarily starve to death.
Late Brezhnev era USSR is an example of bureaucratic despotism.
Conservatism (middle right) is alternative form of stratified society. However, unlike bureaucratic despotism, its closer to feudalism than to caste system. It's very hierarchical with someone above or below someone. Sometimes it lops with an actual unelected monarch on top.
Defenders of this system like to cite various ancient traditions that existed since time immemorial to justify it. They claim that this system existed since forever and therefore proven and tested. They often oppose any change, claiming that it can destabilize society and do harm.
Reality is that most supporters of conservatism are people who already have a warm and affluent status and simply want to shore it up from the rest of the society. The better one is doing in life the more conservative they become. However, using that as argument for a good political system is like when last year champion saying we should not start next season, claiming it can be hectic and chaotic but in reality, he is simply afraid to lose his title to another player and wants to keep it in perpetually.
Revisionist Left (middle left) this is not so much a cohesive system in its own right so much a reaction to failure and devolution of Marxist-Leninist systems into totalitarianism. Many leftists, including even people who worked on creation of USSR, such as Trotskiy, later grew disillusioned and became thinking on safeguards to prevent socialism from devolving into totalitarianism.
Variety of new communist and socialist theories have emerged from that process. All of them combine some elements of socialism with liberal democracy in different proportions. Despised by the far left as traitors and mistrusted by most other ideologies as totalitarian, some of these ideologies are not that bad. Aside from elements of Democratic Socialism and Market Socialism, none were really tried.
Since none were tried, it's hard to estimate how they will playout in practice. However, due to increased automation these ideologies are getting increasingly obsolete and irrelevant to a current technological environment.
Liberal Democracy (middle center) This is a system we currently have. A sort of compromise between all the other sectors. Different parties, movements and such do try to pull law and economy closer towards their sector, but it more or less stays in the middle.
Liberal Democracy has private, public, joint stock, co-op and many other forms of property. Industrial Law regulates balance of power between employees and employers, combining socialism with capitalism into yellow socialism or social capitalism. Various other laws regulate balance of power between individual and state, property owner and tenant, children and parents and so on. A compromise that more or less OK with most even if many would wish to take it further their way.
What is middle in Liberal Democracy is said to be constantly drifting left. The reasons for that are the fact that population grows but numbers of businesses rather shrink than grows, because of that more people are falling into left socio-economic demographic. That however does not mean they will establish even Liberal Socialism but popularity of welfare of Social Democracy will keep growing.
Bottom Row (Libertarianism) Libertarianism is a broad ideology that has many very diverse branches that often very different from each other. However, there is three main ones. Right, left and center. The only thing that unites them is opposition to restrictions, but each of these branches oppose different restrictions.
Propertarianism (bottom right) is right Libertarianism that boils down to property right advocacy. They want to achieve is to let property owners do whatever they want with their property. This is also a branch that complains about taxes most, "taxation" is theft is them. Many of Libertarian right are also nostalgic of the past and religious. They are also the group that advocates various religious freedoms.
All that basically a radical, reactionary or paleo conservatism that is amounts to tyranny of the property owners and powerful. All that creates an illiberal society with arbitrary personal rule and no safeguards for human rights and freedoms.
Overall, I would call far right libertarianism the most misleading of the ideologies. If authoritarian left were simply helpless against the reality of their system, then these people want to create a two-class system of robber baron property owners and peasants who serve them.
Utopian Left (bottom left) This is the most diverse mess of fantasy ideologies, some of which are very different from each other. All sort of utopian well-wishing gathers in this left "libertarian" quadrant. That mixes with anti-revisionist communism, feminism, lifestyle communism, localism, autonomism, kibbutzism and other such ideologies.
Reality that some of this is impossible simply because people will not cooperate with the system. The only real-life example is Rojava's run North-East Syria, that defenders of this quadrant would hail as achievement despite obvious shortcomings. Similar to how they used to hail USSR itself.
That said while Rojava do not seem totalitarian we cannot know for sure, they used to claim the same about USSR itself but now we know the truth. Lib Left still has to reconcile individualism with their rule of the mob, they gave no clear answer to that.
Regardless of any of the above, adherents of this quadrant prefer their utopian objectives to reality. Even if these objectives are backward and irrelevant in current technological reality.
Centre Libertarian (bottom center) Now finally my own quadrant. The only true libertarianism that actually wants to make people free. This is ideology of people who want to pursue happiness and autonomy to achieve that. Free of moral, ideology, religion, excessive regulations and so on.
Geo Libertarianism can actually bridge the gap between rich and poor. Social Libertarianism can give people enough money to live with dignity. Real solutions for real world problems.
Its future prof too, automation already made value of labor worthless in many industries. As time goes, more and more industries would abandon employees in favor of robots. In a system like that people would not be able to earn a living. Social Libertarianism has solutions and that is UBI. Cooperatives would not be able to solve that.
Overall center Libertarian Social Libertarianism is the only true heir to Classical Liberalism that can make people truly free.
That is why Yang says: "not left not right but forward".
