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We propose and demonstrate trapping and rotation of microparticles and biological samples with a moiré-based rotating optical tweezers. We show that polystyrene beads, as well as Escherichia coli cells, can be rotated with ease, while the speed and direction of rotation are fully controllable by a computer, obviating mechanical movement or phase-sensitive interference. Furthermore, we demonstrate experimentally the generation of white-light propelling beams and arrays, and discuss the possibility of optical tweezing and particle micro-manipulation based on incoherent white-light rotating patterns.