Duality invariance implies Poincaré invariance.
We consider all possible dynamical theories which evolve two transverse vector fields out of a three-dimensional Euclidean hyperplane, subject to only two assumptions: (i) the evolution is local in space, and (ii) the theory is invariant under "duality rotations" of the vector fields into one another. The commutators of the Hamiltonian and momentum densities are shown to be necessarily those of the Poincaré group or its zero signature contraction. Space-time structure thus emerges out of the principle of duality.