The origin of pairing in a superconductor resides in the underlying normal state. In the cuprate high-temperature superconductor YBa(2)Cu(3)O(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-x)Eu(0.2)Sr(x)CuO(4) (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.