We investigate a new form of collective dynamics displayed by Thiovulum majus, one of the fastest-swimming bacteria known. Cells spontaneously organize on a surface into a visually striking two-dimensional hexagonal lattice of rotating cells. As each constituent cell rotates its flagella, it creates a tornadolike flow that pulls neighboring cells towards and around it. As cells rotate against their neighbors, they exert forces on one another, causing the crystal to rotate and cells to reorganize. We show how these dynamics arise from hydrodynamic and steric interactions between cells. We derive the equations of motion for a crystal, show that this model explains several aspects of the observed dynamics, and discuss the stability of these active crystals.