The most common and most likely answer to this crossword clue is the 10 letter word POSTMODERN
I've seen this crossword clue in Independent
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More clues leading to the result POSTMODERN
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The straight part
We know that "Perhaps ironic and self-referential" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case perhaps ironic and self-referential = postmodern.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does POSTMODERN mean?
POSTMODERN noun- A postmodernist.
- Of, relating to, or having the characteristics of postmodernism, especially as represented in art, architecture, literature, science, or philosophy that reacts against an earlier modernism.