The origin of pairing in a superconductor resides in the underlying normal state. In the cuprate high-temperature superconductor YBa₂Cu ₃O_y (YBCO), application of a magnetic field to suppress superconductivity reveals a ground state that appears to break the translational symmetry of the lattice, pointing to some density-wave order. Here we use a comparative study of thermoelectric transport in the cuprates YBCO and La _1.8-xEu_0.2Sr_xCuO₄(Eu-LSCO) to show that the two materials exhibit the same process of Fermi-surface reconstruction as a function of temperature and doping. The fact that in Eu-LSCO this reconstruction coexists with spin and charge modulations that break translational symmetry shows that stripe order is the generic non-superconducting ground state of hole-doped cuprates.