Right-handed Dehn twist

In orientation-preserving annular coordinates $(\theta ,r)$ with orientation $d\theta \wedge dr$, the twist is $(\theta ,r)\mapsto (\theta +2\pi f(r),r)$, where $f$ increases from zero to one. The map is the identity outside this annulus and near its boundary. The orientation of the ambient topological surface, rather than the orientation assigned to the curve, fixes the handedness.