The first interstellar object, ‘Oumuamua, was discovered in the solar system by Pan-STARRS in 2017, allowing for a calibration of the abundance of interstellar objects of its size and an estimation of the subset of objects trapped by the Jupiter-Sun system. Photographing or visiting these trapped objects would allow us to learn about the conditions in other planetary systems, relieving the need to send interstellar probes. Here, we explore the orbital properties of captured interstellar objects in the solar system using dynamical simulations of the Jupiter-Sun system and initial conditions drawn from the distribution of relative velocities of stars in the Solar neighborhood. We compare the resulting distributions of orbital elements to those of the most similar population of known asteroids, namely Centaurs, to search for a parameter space in which interstellar objects should dominate and therefore be identifiable solely by their orbits. We find that there should be thousands of ‘Oumuamua-size interstellar objects identifiable by Centaur-like orbits at high inclinations, assuming a number density of ‘Oumuamua-size interstellar objects of ~1015 pc-3. We note eight known objects that may be of interstellar origin. Finally, we estimate that the Large Synoptic Survey Telescope will be able to detect several hundreds of these objects.