That's because authoritarianism isnt an idealology outside of authority worship, it just hijacks the surrounding culture, reclaims it for itself, and transforms all symbols into meaningless team colors. Notice how trump supporters say they represent democracy and freedom despite being opposed to it. Or how christion authoritarians throw around Jesus's name without understanding a word he ever said. Or how Stalinists insist that they're left leaning communists despite cheering for the tanks that crush their comrades. This pattern continues worldwide and affects every nationality and religion.

Authoritarians dont have principles. They have loyalty only, they cannot comprehend non-loyalty principled thinking, and so they see symbols only as a team identifier.

Political compasses dont work for principled worldviews, because principled worldviews cant be graphed on axes. It's just a horoscope effect. Principled worldviews are distinct like pokemon types are. They have to be individually studied to be understood. It is not possible to extrapolate from your first beleif system.