I have often seen this strange hat in medieval art, and wondered at how it was constructed, but recently I came across a passage in a book that explained it. Just as it has been "fashionable" in modern times to wear hats incorrectly, reversed baseball caps, for instance, in the later middle ages rich trendy young men took to rolling up the liripipe hood so that he big collar forms a band , sticking the hole your face peeks out of onto the head, and and draping its long hood around the face. This may well have been the practise of urban gay men as early as the 12th century in Cracow, but that's debated of course, point is it was an affectation initially of a small subset of radical men, but By the 15th century the trend had spread widely and hats were constructed specifically to be worn this way, and could no longer be unrolled back into a hood.