The ocean and the atmosphere, and hence the climate, are governed at large scale by interactions between pressure gradient and Coriolis and buoyancy forces. This leads to a quasigeostrophic balance in which, in a two-dimensional-like fashion, the energy injected by solar radiation, winds, or tides goes to large scales in what is known as an inverse cascade. Yet, except for Ekman friction, energy dissipation and turbulent mixing occur at a small scale implying the formation of such scales associated with breaking of geostrophic dynamics through wave-eddy interactions or frontogenesis, in opposition to the inverse cascade. Can it be both at the same time? We exemplify here this dual behavior of energy with the help of three-dimensional direct numerical simulations of rotating stratified Boussinesq turbulence. We show that efficient small-scale mixing and large-scale coherence develop simultaneously in such geophysical and astrophysical flows, both with constant flux as required by theoretical arguments, thereby clearly resolving the aforementioned contradiction.