A strong volcanic plume consists of a vertical column of hot gases and dust topped with a horizontal `umbrella'. The column rises, buoyed by entrained and heated ambient air, reaches the neutral-buoyancy level, then spreads radially to form the umbrella. In classical models of strong volcanic plumes, the plume is assumed to remain always axisymmetric and non-rotating. Here we show that the updraught of the rising column induces a hydrodynamic effect not addressed to date-a `volcanic mesocyclone'. This volcanic mesocyclone sets the entire plume rotating about its axis, as confirmed by an unprecedented analysis of satellite images from the 1991 eruption of Mount Pinatubo. Destabilized by the rotation, the umbrella loses axial symmetry and becomes lobate in plan view, in accord with satellite records of recent eruptions on Mounts Pinatubo, Manam, Reventador, Okmok, Chaiten and Ruang. The volcanic mesocyclone spawns waterspouts or dust devils, as seen in numerous eruptions, and groups the electric charges about the plume to form the `lightning sheath' that was so prominent in the recent eruption of Mount Chaiten. The concept of a volcanic mesocyclone provides a unified explanation for a disparate set of poorly understood phenomena in strong volcanic plumes.