An experimental and theoretical study of the quadrupole-coupling hyperfine structure of the nonrigid (C2D2)2 dimer is carried out. This dimer exhibits a large amplitude interconversion motion which splits rotational levels into three sublevels. The quadrupole-coupling hyperfine pattern arising from the four deuterium atoms depends on the symmetry species of the tunneling sublevel. For nondegenerate sublevels, the hyperfine structure is especially interesting since the dimer behaves as if the quadrupole coupling were identical for all four deuterium atoms and the effective hyperfine Hamiltonian is completely symmetrical. The symmetry group used to classify the hyperfine levels is, therefore, the permutation group of four objects S4. For the other tunneling sublevels, which are doubly degenerate, the dimer behaves as if the two monomer units were inequivalent. Prior to the diagonalization of the hyperfine Hamiltonian, symmetry-adapted nuclear spin wave functions in S4 are set up and allow us to select hyperfine levels whose symmetry is compatible with the tunneling symmetry species. This formalism is used to analyze the hyperfine patterns of three rovibrational transitions in (C2D2)2, which were recorded under high resolution. The components of effective quadrupole-coupling tensors are thereby determined. These tensors are related to the eQq of an isolated DCCD monomer to obtain vibrationally averaged angles for large amplitude bending motions within the dimer.