One approach, seemingly particularly common (at least on social media), is the one-dimensional left-right paradigm. "Left-wing" and "right-wing" may have several definitions, depending on the context, but on social media discussions it's often not clear what it is meant, and these words are often used to obfuscate meaning and promote propaganda rather than to be informative and clear. Possible definitions:
1: left/right as intended by USA/western culture in late 2010's/early 2020's. Mainly social issues and some economical issues. This seems to be a popular way to categorize it on social media.
2: left/right as purely economical. Marxism to free market capitalism. Relatively more niche than the definition above, and is used in conversations more focused on economics rather than social issues.
3: left/right as intended by USA/western culture in late 2010's/early 2020's. Nazis are far-left. (This seems to be more common among people who self-identify as "right-wing")
4: left/right as intended by USA/western culture in late 2010's/early 2020's. Nazis are far-right. (This seems to be more common among people who self-identify as "left-wing")
5: left/right as purely economical. Marxism-centrism-capitalism-freemarket-nazism. (a bit nonsensical IMHO, but commonly accepted)
6: left/right as purely economical. Marxism-nazism-centrism-capitalism-freemarket.
7: left/right as purely related to the views of the mainstream media at a specific time. Left is mainstream media, right is whatever does not fit within the mainstream media narrative ("right-wing" may be seen as either one ideology or as a collection of different ideologies that merely do not fit within mainstream media narrative, depending on one's own categorization habits). This definition seems to be often used by mainstream media journalists. In this context, "right-wing" may indicate someone who is labelled as a "conspiracy theorist", or someone who promotes "pseudoscience", or someone who is skeptical of climate change (anthropogenic and/or natural), or someone who is against forced vaccinations, or someone who believes in paranormal phenomena or UFO's and so on. This definition is fundamentally different from any other definition listed above and often refers to views that are neither social nor economical, yet it seems to be used frequently in some contexts (like mainstream journalism). Less obfuscating words that one may use may be "pro-estabilishment" and "anti-estabilishment" instead.
8: left/right as purely cultural, defined in the way Twitter users would define it. Same as #1 but completely disregarding economical principles.
9: left/right as change/stability(?) (I've seen this definition quite a few times but personally I find it rather non-sensical)
10: left/right as opposition/affirmation of social/economical hierarchies (a bit vague but makes sense IMHO, especially from a Marxist perspective)
11: right is "liberal" (in the "liberty" sense, so like, pro-LGBT, pro-marijuana, pro-polyamory, pro-guns, pro-abortion, pro-free speech, anti-censorship etc.) whereas left is "strict/conservative/authoritarian", "control over freedom", and stuff like that. Does not fit within the mainstream USA paradigm but I've seen it used by libertarians, who may focus more on consistent principles of "freedom" rather than adopting arbitrary views based on historical contingencies.
12: Same as #11 but switched, so that right is "authoritarian" and left is "liberal/libertarian".
Other views coming!
How to categorize political ideologies
1: One-dimensional left-right paradigm
2: One-dimensional liberal-authoritarian paradigm
3: Two-dimensional left-right liberal-authoritarian paradigm
4: Triangle: liberal+left+right
5: Circle: centrism+left+right+extremism(far-left/far-right)
6: One-dimensional pragmatic-liberal-traditional paragidm
7: Two-dimensional left-right pragmatic-liberal-traditional paradigm
8: Double one-dimensional far-left/left/centrist-left & far-right/right/centrist-right (centrist-left and centrist-right are far from each other)
9: V-Shaped: Far-left/left/centre/right/far-right
10: Inverted V-Shaped: centre-left/left/extremism(far-left&far-right)/right/centre-right
11: One dimensional moderate-extreme paradigm
12: Branched paradigm: many different types of authoritarianism converging into one liberalism
13: Branched paradigm: many different types of right-wing (i.e. anti-mainstream media) converging into one left-wing (mainstream media ideology)
14: Branched paradigm: many different types of pragmatism (with various different goals) converging into one traditionalism (where the goal is maintaining tradition for its own sake)
15: Three-dimensional cube left-right liberal-authoritarian traditional-pragmatic paradigm
16: Various one-dimensional lines of pragmatic-traditional spectrums
17: Various one-dimensional lines of left-right spectrums
18: Various one-dimensional lines of liberal-authoritarian spectrums
19: How that political ideology views violence (e.g. against whom it is justified, for which purposes it is justified, in which cases it is justified etc)
20: How that political ideology views happiness (e.g. who should be happy or deserves to be happy, through what kind of means one should or is allowed to attain happiness, what sources of happiness are allowed, its views on hedonism/asceticism etc)
21: How that political ideology views hugs (this may at least partly indicate what kind of cultural/social views this ideology has)
One-dimensional spectrums (line):
1: left/right
2: pragmatic/idealistic/traditional (pragmatic: the result is prioritized, idealistic: an ideal is prioritized, traditional: tradition is prioritized)
3: liberal/authoritarian
4: who benefits? self(individual)/other(individual)/community(that one identifies as part of)/community(that one does not identify as part of)/the rulers in charge/a particular deity
5: anti-violence/pro-violence
6: selfishness/altruism
7: BSD-like freedom/GNU-like freedom (BSD-like freedom means, for example, that a company can apply restrictive rules, e.g. a big tech social media can censor any speech they don't like from their platform, and is therefore more in line with mainstream libertarianism, while GNU-like freedom would disallow that).
8: anarchy (i.e. lack of hierarchies)-extreme hierarchies
Two-dimensional spectrums (square):
1: x: left-right, y: liberal-authoritarian
2: x: pragmatic/idealistic/traditional, y: liberal-authoritarian (i.e. in this case, it's mostly about how much people can deviate from the officially sanctioned ideology/behavior)
3: x: social equality-inequality, y: liberal-authoritarian (i.e. in this case, it's mostly about the prevalence of people who can deviate significantly from the officially sanctioned ideology/behavior)
4: x: amount of hierarchy, y: what the people who are superior in the hierarchy can do to the people below
Three-dimensional spectrums (cube):
1: x: left-right, y: (legally, not culturally) liberal-authoritarian, z: culturally (not legally) liberal-authoritarian
Four-dimensional spectrums (tesseract):
Definitions:
Left/right:
1: left/right as intended by USA/western culture in late 2010's/early 2020's. Mainly social issues and some economical issues.
2: left/right as purely economical. Marxism to free market capitalism.
3: left/right as intended by USA/western culture in late 2010's/early 2020's. Nazis are far-left.
4: left/right as intended by USA/western culture in late 2010's/early 2020's. Nazis are far-right.
5: left/right as purely economical. Marxism-centrism-capitalism-freemarket-nazism. (a bit nonsensical IMHO, but commonly accepted)
6: left/right as purely economical. Marxism-nazism-centrism-capitalism-freemarket.
7: left/right as purely related to the views of the mainstream media at a specific time. Left is mainstream media, right is whatever does not fit within the mainstream media narrative ("right-wing" may be seen as one ideology or as a collection of different ideologies that merely do not fit within mainstream media narrative, depending on one's own categorization habits).
8: left/right as purely cultural, defined in the way Twitter users would define it.
9: left/right as change/stability(?)
10: left/right as opposition/affirmation of social/economical hierarchies
11: left is "liberal" (in the "liberty" sense, so like, pro-LGBT, pro-marijuana, pro-polyamory, pro-guns, pro-abortion, pro-free speech, anti-censorship etc.) whereas right is "strict", "control over freedom", and stuff like that. Does not fit within the contemporary USA paradigm.