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Polarization-mode dispersion (PMD) causes significant impairment for high bit-rate optical telecommunications systems. It is known that PMD can be strongly reduced by spinning the fiber as it is drawn. In this paper, we focus on the case of randomly birefringent fibers spun at a constant rate, providing analytical expressions for the asymptotic statistical properties of PMD. In particular, we investigate the behavior of the first- and second-order PMD, demonstrating that a constantly spun fiber behaves asymptotically as an unspun fiber. Conversely, we show that the distance at which the PMD reaches its asymptotic trend increases with the spin rate up to lengths of several kilometers.