Spin-orbit coupling emerging out of a material's global inversion symmetry breaking has been long known to act as a pair breaking mechanism to the superconducting order parameter. However, when spin-orbit coupling emerges from local inversion symmetry breaking, an unexpected coexistence with superconductivity might occur. Although lattice driven local symmetry breaking is intrinsic to a variety of unconventional superconductors, little is known about its close connection to spin-orbit coupling, and how the combination of the two can impact electronic properties and the microscopic classification of the order parameter. By using high-resolution spin- and angle-resolved photoemission spectroscopy we reveal the presence of a universal momentum-dependent spin texture throughout the hole-doped side of the cuprate phase diagram for the Bi-based family, which becomes negligible on the electron-doped side. We attribute this spin asymmetry to local inversion symmetry breaking induced by local distortions of the CuO octahedra, revealing an interplay between spin-orbit coupling and lattice symmetry. The implication of this connection in steering the ground state properties of cuprate superconductors is discussed, together with the intriguing possibility that it might provide the long missing link to explain the complexity and asymmetry of the cuprate phase diagram.