Vorticity generation in accretion disks around Schwarzschild and Kerr black holes is investigated in the context of magnetofluid dynamics derived for both General Relativity (GR) and modified gravity formulations. In both cases, the Kerr geometry leads to a "stronger" generation of vorticity than its Schwarzschild counterpart. Of the two principal sources, the relativistic drive peaks near the innermost stable circular orbit (isco), whereas the baroclinic drive dominates at larger distances. Consequences of this new relativistic vorticity source are discussed in several astrophysical settings. © 2015 American Physical Society.