Bacterial motility involves switching between the left and right supercoiled states of the flagellar filament. The polymorphism of this assembly of identical flagellin molecules has presented a structural puzzle. Supercoiling has been attributed to coexistence of two conformational states of the 11 nearly axially aligned protofilament strands of subunits. The helical parameters of straight filaments in the left (L) and right (R) lattice states have now been accurately determined by X-ray fiber diffraction. The 9 A resolution electron density map of the R-type filament, refined from the X-ray data, reveals the interlocked alpha-helical segments of the core portion, which constitute the inner and outer tubes. While the inner-tube domain interactions remain invariant, the strand joints in the outer tube can switch between the L- and R-state by 2-3 A axial shifts, which change the strand periodicity of approximately 50 A by 0.8 A. This bi-stable quaternary switching results in supercoiling. Based on the measured helical parameters of the L and R lattices and the switching model, the twist and curvature calculated for the ten possible supercoils are in quantitative accord with observed supercoiled forms of flagellar filaments.