  Under continuity and recurrence assumptions, we prove that the iteration of
successive partial symmetrizations that form a time-homogeneous Markov process,
converges to a symmetrization. We cover several settings, including the
approximation of the spherical nonincreasing rearrangement by Steiner
symmetrizations, polarizations and cap symmetrizations. A key tool in our
analysis is a quantitative measure of the asymmetry.
