Centrifugal buoyancy affects all rotating turbulent convection phenomena, but is conventionally ignored in rotating convection studies. Here, we include centrifugal buoyancy to investigate what we call Coriolis-centrifugal convection (C^{3}), characterizing two so far unexplored regimes, one where the flow is in quasicyclostrophic balance (QC regime) and another where the flow is in a triple balance between pressure gradient, Coriolis and centrifugal buoyancy forces (CC regime). The transition to centrifugally dominated dynamics occurs when the Froude number Fr equals the radius-to-height aspect ratio γ. Hence, turbulent convection experiments with small γ may encounter centrifugal effects at lower Fr than traditionally expected. Further, we show analytically that the direct effect of centrifugal buoyancy yields a reduction of the Nusselt number Nu. However, indirectly, it can cause a simultaneous increase of the viscous dissipation and thereby Nu through a change of the flow morphology. These direct and indirect effects yield a net Nu suppression in the CC regime and a net Nu enhancement in the QC regime. In addition, we demonstrate that C^{3} may provide a simplified, yet self-consistent, model system for tornadoes, hurricanes, and typhoons.