The imaginary Poynting momentum (IPM) of light has been captivated as an unusual origin of optical forces. However, the IPM force is predicted only for dipolar magnetoelectric particles that are hardly used in optical manipulation experiments. Here, we report a whole family of high-order IPM forces for not only magnetoelectric but also generic Mie particles, assisted with their excited higher multipoles within. Such optomechanical manifestations derive from a nonlocal contribution of the IPM to the optical force, which can be remarkable even when the incident IPM is small. We observe the high-order optomechanics in a structured light beam, which, despite carrying no angular momentum, is able to set normal microparticles into continuous rotation. Our results provide unambiguous evidence of the ponderomotive nature of the IPM, expand the classification of optical forces, and open new possibilities for levitated optomechanics and micromanipulations.