The origin of high-temperature superconductivity in copper oxides and the nature of the 'normal' state above the critical temperature are widely debated(1-3). In underdoped copper oxides, this normal state hosts a pseudogap and other anomalous features; and in the overdoped materials, the standard Bardeen-Cooper-Schrieffer description fails(4), challenging the idea that the normal state is a simple Fermi liquid. To investigate these questions, we have studied the behaviour of single-crystal La2-xSrxCuO4 films through which an electrical current is being passed. Here we report that a spontaneous voltage develops across the sample, transverse (orthogonal) to the electrical current. The dependence of this voltage on probe current, temperature, in-plane device orientation and doping shows that this behaviour is intrinsic, substantial, robust and present over a broad range of temperature and doping. If the current direction is rotated in-plane by an angle phi, the transverse voltage oscillates as sin(2 phi), breaking the four-fold rotational symmetry of the crystal. The amplitude of the oscillations is strongly peaked near the critical temperature for superconductivity and decreases with increasing doping. We find that these phenomena are manifestations of unexpected in-plane anisotropy in the electronic transport. The films are very thin and epitaxially constrained to be tetragonal (that is, with four-fold symmetry), so one expects a constant resistivity and zero transverse voltage, for every phi. The origin of this anisotropy is purely electronic-the so-called electronic nematicity. Unusually, the nematic director is not aligned with the crystal axes, unless a substantial orthorhombic distortion is imposed. The fact that this anisotropy occurs in a material that exhibits high-temperature superconductivity may not be a coincidence.