A constantly rotating spherical container sustains turbulence of a fluid partially filling it. This simple turbulence generator has the potential for wide engineering applications as a bladeless mixer. Using the coupled level-set and volume of fluid method and the boundary data immersion method, we conduct direct numerical simulations of liquid–gas flow in a spherical container rotating about a horizontal axis to investigate the driving mechanism of turbulence, flow dependence on the filling rate $\varPsi$ and the mixing ability of the sustained turbulence. Even if the Froude number $Fr$ is small enough ($Fr\lesssim 10^{-3}$) for the liquid–gas interface to be undeformed, if the Reynolds number $Re$ is large enough ($Re\gtrsim 10^3$), small-scale turbulent eddies are sustained by being stretched in shear flow around a counter-rotating pair of container-size vortices, whose swirling directions depend on $\varPsi$. We clarify that the angle of flow near the solid wall colliding with the interface controls the swirling direction of these container-size vortices. Furthermore, we track fluid particles in the liquid phase to quantify mixing properties to show that almost perfect mixing occurs after approximately 10 spins of the container for lower $\varPsi$ ($\lesssim 0.5$), whereas the mixing requires less energy consumption for higher $\varPsi$ ($\gtrsim 0.7$) at the examined $Re=O(10^3)$.