I politely suggest that the reader imagine an American football. It has stitches down the middle, dividing the ball into hemispheroids, which we will call "left" and "right". Stand the ball on one end, so that the stitches run vertically. Note that the stitches do not run the length of the ball. Instead, there are two regions, one at each pole, where there is no stitching. We will call these "danger zones".
This football is a map of how societies form and operate, and positions on the football will correspond to political ideologies. Imagine that the vertical axis, from the bottom to the top, is a measure of how cooperative people are towards each other. At the bottom, people do not cooperate except for the bare minimum necessary to survive and reproduce. At the top, people are completely eusocial and the interests of society replace the interests of the individual. Lateral movement on the ball corresponds to a spectrum from conservatism to modernism, right to left.
What is cooperation? For us, it will be about the willingness to participate in centralized power structures, where people agree to exchange some of their own personal power for some security or guarantee from a centralized power. Note that this is a very generic and flexible idea, allowing for people to exchange anything from money to food, to land, to fame, to social connections, to reproductive rights.
What do various lateral positions look like? Lateral positions are always relative to a cultural context. Culture is the societal medium along which memes are propagated. We can imagine the football slowly rotating in the same direction as the Earth, moving memes from new, progressive or modernist positions across the center to become social norms which are then preserved by conservative or traditionalist positions. Note that footballs are spheroid, and so cultural trends and memetic signals tend to repeat. This analogy is not perfect, but only because we do not yet understand what causes the repetition of patterns throughout history.
What do various vertical positions look like? People cluster into families, tribes, communities, cities, and states. The amount of clustering possible is recursively dependent on the number of power structures and their relative strengths, like a weighted Voronoi diagram.
The stitches and danger zones themselves are relatively continuous, but they are also connected by relatively sharp phase transitions. We will therefore examine each region individually, and then consider the transitions between them.
In the southern danger zone, states effectively do not exist. People will not work together with sufficient altruism and social empathy to develop a state powerful enough to ensure justice or distribute essential goods. While individual cities may exist, and nation-states may declare themselves to have sovereignty over tracts of land, it is easy for a band of motivated people to resist the will of any state upon them.
In the northern danger zone, states are more powerful than any other collective of people, and people are eusocially reduced to cogs in the machine, ants in the ant colony, or bees in the hive. The state demands all personal rights and gives nothing in return.
In the stitches, a variety of states are possible. People routinely exchange nearly all of their personal rights for state-granted rights. People can organize alongside the state, lobbying, unionizing, corporatizing, and rebelling. These collective actions compete with the state, leading to tension and tensegrity.
The lower phase transition up into the stitching is sharp because it involves the binary question of whether a typical individual is able to materially ignore the will of the state. The upper phase transition is less sharp; although it should be clear when a fascist state is the most powerful structure acting on a population, it seems that fascist states are brittle in practice.
We can imagine most vertical political positions as a measurement of the strength of the state, ideologically, but positions within the danger zones are hostile to people and society as we know and enjoy it. All states are hostile towards people at some level, since states are self-interested, but states also must care for people, or else they will run out of power. Thus it would seem that we should desire to reside within the stitches.
The crucial insight delivered by the football metaphor now becomes apparent. Within the stitching, the possibilities for social expression stretch far and wide. There are many possible states and thus many possible societies, each with differing styles of governance, markets, rights, and cultural tendencies. It becomes possible to have far-left positions where markets are wholly state-run, as well as far-right positions where corporations write legislation.
However, at the southern danger zone, the state becomes too weak to exist at a scale which can support state-run markets. Indeed, states simply cannot get very large in the zone, so corporations themselves cannot exist, since they rely on state bureaucracy for deriving the bulk of their power. People are thus reduced to loose federation and tribes, with cities being possible in some situations. We can see that anarcho-capitalists and anarcho-communists alike must adopt many similar behaviors in order to survive: Water, food, weaponry, medicine, and other essentials will not be provided by any central power. While people certainly may choose to behave differently in an anarchist situation, the fact is that all anarchist worlds are extremely similar.
At the northern danger zone, the state becomes stronger than the combined will of the people to resist it, and forms a fascist state. The metaphor of the fasces is still useful for explaining what this is like: A fasces is a bundle of sticks, and a fascist state is a bundle of people. The bundle is strong, but sometimes individual sticks break and must be replaced; a fascist state consumes people, using them for its own ends. Once this happens, any interesting distinctions between lateral positions evaporate, because we shift from a position of people to a position of states. A far-left state in the zone and a far-right state in the zone are both fully capable of manipulating markets throughout the realm, regardless of the precise bureaucratic mechanism. Similarly, the rights of people are no longer respected, because they are unimportant to the state, so neither progressive nor traditionalist agendae can advance. Finally, the claimed ideology of the state does not matter, because the state's actions are not ideologically influenced; instead, ideological claims are mere words to be used against people who cannot be moved with power alone.
On a historical note, it has often been the case that fascists will play on left-versus-right feelings in order to assign more power to themselves. For this reason, it is imperative to understand that fascism prefers the state in and of itself, and not any particular goals, means, or ends beyond the state.
It seems to be the case that, just like the football seems to turn, it also seems to have an upward motion of some sort, at least historically. As remarked earlier, states are self-interested once they become self-supporting. It would thus make sense that states are always striving to strengthen themselves and achieve fascism. This would also be a strong argument in favor of democratic designs which empower the people to control the state.
The political football provides a better intuition than the political compass, by providing fewer corners and corresponding those corners more closely to states and social behavior.