The emerging field of structured beams has led to optical manipulation with tremendous progress. Beyond various methods for structured beams, we use phase-shifted zone plates known as beam-shaping diffractive optical elements to generate beams whose phase exclusively or both phase and intensity are twisted along a curve. These beams can trap and guide particles on open curved trajectories for continuous motion, not necessarily requiring a closed symmetric intensity distribution. We show the feasibility and versatility of the proposed method as a promising technique in optical manipulation in which the trajectory of the spiral rotation and the rate of rotation of trapped particles can be controlled.