The ionized rare gas clusters Ar-4(+) and Xe-4(+) have been studied by photofragmentation. Center-of-mass velocities of the neutral and ionized photofragments have been measured for photon energies between 1.6 and 4.4 eV. In almost all of the examined photon energy region, both clusters exhibit fragment velocity distributions which show that they consist of a linear ionized trimer with an additional atom loosely attached to its side. For the lowest photon energies, however, fragmentation patterns are observed which can only be explained by a linear tetramer structure. This demonstrates that stable linear isomers of the ionized rare gas tetramers exist.