A complicating factor in unravelling the theory of high-temperature (high-T-c) superconductivity is the presence of a 'pseudogap' in the density of states, the origin of which has been debated since its discovery(1). Some believe the pseudogap is a broken symmetry state distinct from superconductivity(2-4), whereas others believe it arises from short-range correlations without symmetry breaking(5,6). A number of broken symmetries have been imaged and identified with the pseudogap state(7,8), but it remains crucial to disentangle any electronic symmetry breaking from the pre-existing structural symmetry of the crystal. We use scanning tunnelling microscopy to observe an orthorhombic structural distortion across the cuprate superconducting Bi2Sr2Can-1CunO2n+4+x (BSCCO) family tree, which breaks two-dimensional inversion symmetry in the surface BiO layer. Although this inversion-symmetry-breaking structure can impact electronic measurements, we show from its insensitivity to temperature, magnetic field and doping, that it cannot be the long-sought pseudogap state. To detect this picometre-scale variation in lattice structure, we have implemented a new algorithm that will serve as a powerful tool in the search for broken symmetry electronic states in cuprates, as well as in other materials.