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.