I would argue that we judge a community or system of thought or a person as rational or not based in large part upon how they respond to criticism. If the response to criticism is to ignore it or shut it up by appeal to some authority — a text, a leader, “the way we do things” — it is less rational. If the response is to engage in dialogue with defeasible appeals to public criteria it is more rational. No community or person is completely rational or irrational. Even the homicidal maniac ax murderer will listen if you tell him that his ax handle is falling apart. Even the most secular atheist utilitarian will sometimes make an appeal to life or happiness or freedom which he or she cannot further justify. However just because there is no pure example of a rational or irrational person or group does not mean it is not possible to detect tendencies and groupings — the Manson Family is less rational than the US Supreme Court for example.
Explicitly anti-rational ideologies — romanticism, mysticism, fascism — are an interesting test case. They provide an argument against appealing to argument — it makes life boring, or vitiates the sacred bonds of the community, or teaches children to be uppity or whatever. By doing so they are appealing to a reason not to be rational. Nothing wrong with that, but it demonstrates how in a modern society some sort of appeal to what’s right as opposed to “just how you have to do things” is inescapable.
The connection between rationality and a non-brow-beating response to criticism also demonstrates how intertwined issues of rationality are with issues of ethics. Part of the reason you don’t want to be irrational is that you don’t want to get things wrong about the world — if you are stuck in a conceptual rut, or factually in error, or trapped at a local maximum you would like to know. But another part of the reason is that knocking down everyone who challenges you with an appeal to authority is just an awful way to be a person.