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Vortex beams (VBs) carrying orbital angular moment (OAM) modes have been proven to be promising resources for increasing communication capacity. Although considerable attention has been paid on metasurface-based VB generators due to the unprecedented advantages of metasurface, most applications are usually limited at a single band with a fixed OAM mode. In this work, an emerging dual-band reflection-type coding metasurface is proposed to mitigate these issues by newly engineered meta-atoms, which could achieve independent 2-bit phase modulations at two frequency bands. The proposed coding metasurface could efficiently realize and fully control dual-band VBs carrying frequency selective OAM modes under the linearly polarized incidence. As the first illustrative example, a dual-band VB generator with normal beam direction is fabricated and characterized at two widely used communication bands (Ku and Ka bands). Moreover, by encoding proper coding sequences, versatile beams carrying frequency selective OAM modes can be achieved. Therefore, by adding a gradient phase sequence to the first VB generator, the second one is designed to steer the generated beams to a preset direction, which could enable diverse scenarios. The measurement results of both VB generators agree very well with the numerical ones, validating the full control capability of the proposed approach.