Here is another part of the parent's post which I alluded to:
> the people on the American political right see it as what this website describes it as “ A self-righteously moralistic person who behaves as if superior to others.”
This is why it's ironic. Two sides of the same coin: one group of righteous people claims moral superiority over another group of infidels, and vice-versa.
It's the Spidermen pointing at themselves meme. These conflicting beliefs can reasonably co-exist (as they always have done), so who stands to gain from treating them as mutually exclusive (there can only be one?)?