ανασηκώνει λίγο την πλατύγυρη ρεπούμπλικα προς τα πίσω

Think how such a movement might be described in English.  If you have access to the BNC or the Bank of English, try searching on the words "hat back" and see what you can get.  [for the Bank of English, put a + between the words, rather than a space.]   Notice that "raised his hat" would mean took it off his head, which is not what he did.

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