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Turbulence is ever produced in the low-viscosity/large-scale fluid flows by the velocity shears and, in unstable stratification, by buoyancy forces. It is commonly believed that both mechanisms produce the same type of chaotic motions, namely, the eddies breaking down into smaller ones and producing direct cascade of turbulent kinetic energy and other properties from large to small scales towards viscous dissipation. The conventional theory based on this vision yields a plausible picture of vertical mixing and remains in use since the middle of the 20th century in spite of increasing evidence of the fallacy of almost all other predictions. This paper reveals that in fact buoyancy produces chaotic vertical plumes, merging into larger ones and producing an inverse cascade towards their conversion into the self-organized regular motions. Herein, the velocity shears produce usual eddies spreading in all directions and making the direct cascade. This new paradigm is demonstrated and proved empirically; so, the paper launches a comprehensive revision of the theory of unstably stratified turbulence and its numerous geophysical or astrophysical applications.