High-temperature superconductivity in the cuprates emerges from an enigmatic metallic state, known as the pseudogap, characterized by a reconstructed Fermi surface, reduced carrier density, and the appearance of Fermi arcs, whose origin remains unresolved. Here, we show that these defining signatures naturally arise from a structural reconstruction observed experimentally that introduces a symmetry-enforced sublattice degree of freedom. In the presence of spin-orbit coupling, the Fermi surface is reconstructed into small closed pockets, effectively reducing the carrier density. The same sublattice structure gives rise to matrix-element interference in angle-resolved photoemission spectroscopy, leading to the manifestation of Fermi arcs. Density functional theory calculations support this mechanism. These results demonstrate that lattice symmetry provides a unifying and experimentally verifiable framework for understanding the pseudogap regime in the cuprates.