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Spun fibers are increasingly used in telecommunication systems because their polarization-mode dispersion (PMD) is lower than that in unspun fibers. In this paper, we investigate the effects of a periodic spin on the PMD of fibers with randomly varying birefringence. Numerical simulations show that when the spin period is of the same order as or larger than the beat length, the mean differential group delay of a spun fiber depends on the model used for the random birefringence. We then carry out a general theoretical analysis using the second Wai-Menyuk model, which is the only model of fiber birefringence to date that is consistent with polarization optical time domain reflectometry data. Finally, we consider some particular regimes by means of a perturbative approach.