Ants On A Circle I

Nine ants are placed at equal spacing around a circle. Each ant independently chooses clockwise or counterclockwise and then moves at constant speed so that, if uninterrupted, each would make exactly one full revolution in one minute. When two ants meet they instantly reverse direction and continue at the same speed. The ants are indistinguishable, so we only care about which geometric points around the circle are occupied, not which ant is which. What is the probability that after one minute the set of occupied points is exactly the same set of nine starting points?