I'd say they do. Look at the number of conservative groups that sprang up against Trump. I haven't heard of any left-leaning groups forming against the extreme end of the "culture war" that leans left, e.g. the one that has imposed rules on ChatGPT that make it give this absurd (and entirely unethical by any reasonable ethics framework) answer. The only thing I've seen get pushback in left-leaning circles is advocating for actual communism in the economic sense.
Haven't you ever heard of the "Dirtbag left"? Their podcasts are extremely popular and they can hardly be accused of being "woke". WSWS is an example of a very hard-socialist site.
Will Shetterly is often credited with inventing the term "Social Justice Warrior". He called it identitarianism, made a decent case that it first appeared in Unitarian-Universalist circles in the 90s, and has been criticizing it ten years before anyone else. He's also a communist and a Civil Rights veteran.
As much as actual communists fight among themselves, it'd be a miracle if the tendency of "identitatian deference" hadn't been identified and criticized by someone long before the right noticed it.
I don't see much pushback on either side. Liberals see themselves as champions of freedom and are quite convinced that conservatives are by and large fascist sympathizers. In this regard, both sides strongly believe themselves to have the moral high road, and all nuance has been tossed aside. Interestingly, I find that both sides are quite capable of making very plausible, reasoned arguments that make them sound like the adults in the room.
I think that to a surprising extent, liberals and conservatives live in different versions of America. I try to keep my news exposure to a minimum, but I do make a point of visiting the major outlets for both ideologies. You can really see how reality itself is curated. I don't even think it's really curated ideologically, either, my feeling is that media outlets know their audience and make sure they are reinforcing the relationship instead of challenging it.
