We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra H along with a distinguished element t ∈ H such that (H, R, C) is a ribbon Hopf algebra, where R = (t −1 ⊗ t −1 ) (t) and C = t −2 . The element t is closely related to the topological "half-twist", which twists a ribbon by 180 degrees. We construct a functor from a topological category of ribbons with half-twists to the category of representations of any half-ribbon Hopf algebra. We show that U q (g) is a (topological) half-ribbon Hopf algebra, but that t −2 is not the standard ribbon element. For U q (sl 2 ), we show that there is no half-ribbon element t such that t −2 is the standard ribbon element. We then discuss how ribbon elements can be modified, and some consequences of these modifications.