So I have a question followed by a few notes:
What are the ideological colors for lack of a better term? Socialism OTL uses red as a consistent example. It's mentioned that Eduism favors green and black seems like a safe bet for Beutelism, but what about the rest? The fascists as an American led bloc most likely use Union Blue instead of OTL brown, but I can't remember if I read that Illuminism uses red or if I made that up. Maybe yellow representing light? I know Europa is blue on the maps but I don't think of "authoritarian monarchism" as having a strong color association (besides perhaps purple).
If the Illuminist Bloc becomes Social Capitalist that opens up a tripartite economic struggle in the twentieth century between corporatism, capitalism, and syndicalism, and would lead to *American politicians denouncing capitalism as a dire threat to the American way of life while glorifying corporate power.
This next one's a doozy. Going back to the faction color question, given Evola's OTL belief in alchemy and likely desire to create a sort of Brave New World caste system TTL, I think that using the steps of the alchemical Magnum Opus as caste names would make sense. The first step (Nigredo) represents disorganized chaos and symbolizes the lower classes, using the sonnenrad as a symbol. The next step (Albedo) uses a crescent moon symbol and represents purification. It could represent whatever "middle class" means in Europa, or the business class or whatever. The third step (Citrinitas), symbolized by a dot in a circle representing the sun, represents enlightenment and could consist of the religious/government/military bureaucracy. The final step (Rubedo), symbolized by a combination of the sun and moon symbols, represents perfection and the aristocracy/leader caste.
It's not perfect I'll admit but attempting to impose some sort of societist caste system on a twentieth century great power was never going to track perfectly anyway. The end result would likely be a tiered pyramid symbol, perhaps ending at the yellow Citrinitas stage with the Rubedo symbol floating above it like an unholy Eye of Providence? You'd end up with iconography vaguely similar to the "Five Races One Country" concept in the Republic of China, albeit obviously ranked, and would create an amusing scenario where a rainbow symbol represents a repressive traditionalist totalitarian ideology. There would even be a name for the arrangement, as the Cauda Pavonis in alchemy is represented by a spectrum of colors.