Sudden single-nucleon removal reactions from fast radioactive beams are now key to studies of the structure of rare isotopes. The sensitivity of the heavy residue's parallel momentum distribution to the orbital angular momentum of the removed nucleon is a crucial feature with a high spectroscopic value. Twonucleon removal reactions provide experimental reach toward the rarest nuclear species. We show that the residue parallel momentum distributions in these reactions offer a clear spectroscopic signal of the angular momentum of the pair of nucleons removed, and thus of the residue final state spins and spectroscopy. Our formalism is applied successfully to new final-state-inclusive measurements of like-nucleon pair removal reactions to states in neutron-rich 36 Mg and neutron-deficient 20 Mg. We also confront a new final-stateexclusive decomposition of two-proton knockout data to states in neutron-rich 26 Ne. DOI: 10.1103/PhysRevLett.102.132502 PACS numbers: 23.20.Lv, 21.60.Cs, 24.50.+g, 27.30.+t Intermediate-energy one-nucleon knockout reactions from rare-isotope beams are now key to our obtaining spectroscopic information on the dominant proton and neutron single-particle structures near the Fermi surfaces of short-lived isotopes far from the valley of stability [1,2]. The potency of the method, aside from its high experimental efficiency, is the combination of measurements of cross sections and their distribution with the parallel momentum of the projectilelike residues that derive information on the spectroscopic strength and orbital angular momentum of the removed nucleon, respectively. Analyses have been used to deduce quantitative singleparticle spectroscopy in many instances, e.g., [3][4][5][6], and contributed to our understanding of the evolution of shell structures in nuclei with the most extreme N:Z ratios. Recent examples include [7][8][9][10].We consider analogous two-nucleon removal reactions, events in which intermediate-energy, mass A projectiles undergo grazing collisions with a light nuclear target, resulting in the sudden removal of two like nucleons. The reactions of interest involve the removal of two protons from neutron-rich and two neutrons from neutron-deficient projectiles. In both cases the removed nucleons are strongly bound. We describe measurements where only the mass A À 2 projectile residues (denoted c) are detected. Ideally, the energy of the final states of the residues are also identified by measurement of their in-flight decay rays. Key to physical interpretation is that, in the sudden approximation, the sum,~1 þ~2, of the momenta of the removed nucleons in the projectile rest frame is probed by the laboratory momenta of the projectileK A and the residueK c measured in two-nucleon removal events,~1shown schematically in Fig. 1. Experiments determine the residue momentum component relative to a single direction. While early work on halo nuclei [11] looked at the component transverse to the beam direction, the distributions with respect to the parallel component are now pre...