We investigate the dynamics of mobile inclusions embedded in 2D active nematics. The interplay between the inclusion shape, boundary-induced nematic order, and autonomous flows powers the inclusion motion. Disks and achiral gears exhibit unbiased rotational motion, but with distinct dynamics. In comparison, chiral gear-shaped inclusions exhibit long-term rectified rotation, which is correlated with dynamics and polarization of nearby +1/2 topological defects. The chirality of defect polarities and the active nematic texture around the inclusion correlate with the inclusion's instantaneous rotation rate. Inclusions provide a promising tool for probing the rheological properties of active nematics and extracting ordered motion from their inherently chaotic motion.