Kinda hard to be authoritarian if one of your central beliefs is in the fraternity and equality of all. It's more useful to describe Stalinist USSR or Maoist China as right-wing regimes cloaked in leftist rhetoric.
This sums it up pretty well: "Left-wing authoritarians, in Costello's research, typically strongly agree with the following statements: the rich should be stripped of their belongs and status; deep-down just about all conservatives are racist, sexist and homophobic; classrooms can be safe spaces that protect students from the discussion of harmful ideas." (https://www.newstatesman.com/international/2020/05/there-suc...)
All of which lines up with the rhetoric from the Bolsheviks and the Maoists and a lot of the modern day critical race theorists.
Critical race theory has more in common with National Socialism than with the leftist values of equality and solidarity, because when you get past the verbal subterfuge which ostensibly denies that race exists, you find a deeply racist ideology of racial collectives locked in a struggle for supremacy, with one particular collective singled out as evil beyond redemption and in need of destruction. We have seen that before, and it is undeniable where it must lead. Critical Race Theory is Nazism barely disguised by a light sprinkling of leftist pixie-dust.
The common factor between the extremes of left and right is collectivism. Totalitarian demagogues, left or right, favor collectivism as a tool of control because it detaches people from their moral obligations to other individual human beings and transfers their loyalties wholly to an abstraction controlled by that demagogue.
Authoritarians are invariably toxic damaged people who are full of rage and are looking for a target to make themselves feel better.
Sometimes they end up on the left. Sometimes they end up on the right. Sometimes they end up in management or venture capital.
Same issues, same dynamic, different context.
The difference is authoritarians on the right are more likely to use physical violence. "Cultural" authoritarians are more likely to use social exclusion - brigading on social media, getting people fired, and so on.
The key point is that the presenting issue is irrelevant. The social dynamic and mode of relationship - angry tribalism used to justify violence without any sense of context - is the real tell.