How turbulent convective fluctuations organize to form larger-scale structures in planetary atmospheres remains a question that eludes quantitative answers. The assumption that this process is the result of an inverse cascade was suggested half a century ago in two-dimensional fluids, but its applicability to atmospheric and oceanic flows remains heavily debated, hampering our understanding of the energy balance in planetary systems. We show using direct numerical simulations with spatial resolutions of 122882 × 384 points that rotating and stratified flows can support a bidirectional cascade of energy, in three dimensions, with a ratio of Rossby to Froude numbers comparable to that of Earth's atmosphere. Our results establish that, in dry atmospheres, spontaneous order can arise through an inverse cascade to the largest spatial scales.