This article analyzes the behaviour of helicoidal ends of properly embedded minimal surfaces, namely properly embedded infinite total curvature minimal annuli of parabolic type, satisfying a growth condition on the curvature via the Gauss map, and a geometric transversality condition. Then we show that embeddedness forces the end to be asymptotic either to a plane, or a helicoid or a spiraling helicoid. In all three cases, the Gauss map can be described in very simple terms. Finally this local result yields a global corollary stating the rigidity of embedded minimal helicoids.