Non-Abelian flux-tube (string) solutions carrying global currents are found in the bosonic sector of 4-dimensional N=2 super-symmetric gauge theories. The specific model considered here posseses U(2)local x SU(2)global symmetry, with two scalar doublets in the fundamental representation of SU(2). We construct string solutions that are stationary and translationally symmetric along the x3 direction, and they are characterized by a matrix phase between the two doublets, referred to as "twist". Consequently, twisted strings have nonzero (global) charge, momentum, and in some cases even angular momentum per unit length. The planar cross section of a twisted string corresponds to a rotationally symmetric, charged non-Abelian vortex, satisfying 1st order Bogomolny-type equations and 2nd order Gauss-constraints. Interestingly, depending on the nature of the matrix phase, some of these solutions even break rotational symmetry in R3. Although twisted vortices have higher energy than the untwisted ones, they are expected to be linearly stable since one can maintain their charge (or twist) fixed with respect to small perturbations.