Rotation-powered millisecond radio pulsars have been spun up to their present spin period by a $10^8$ - $10^9$ yr long X-ray-bright phase of accretion of matter and angular momentum in a low-to-intermediate mass binary system. Recently, the discovery of transitional pulsars that alternate cyclically between accretion and rotation-powered states on time scales of a few years or shorter, has demonstrated this evolutionary scenario. Here, we present a thorough statistical analysis of the spin distributions of the various classes of millisecond pulsars to assess the evolution of their spin period between the different stages. Accreting sources that showed oscillations exclusively during thermonuclear type I X-ray bursts (nuclear-powered millisecond pulsars) are found to be significantly faster than rotation-powered sources, while accreting sources that possess a magnetosphere and show coherent pulsations (accreting millisecond pulsars) are not. On the other hand, if accreting millisecond pulsars and eclipsing rotation-powered millisecond pulsars form a common class of transitional pulsars, these are shown to have a spin distribution intermediate between the faster nuclear-powered millisecond pulsars and the slower non-eclipsing rotation-powered millisecond pulsars. We interpret these findings in terms of a spin-down due to the decreasing mass-accretion rate during the latest stages of the accretion phase, and in terms of the different orbital evolutionary channels mapped by the various classes of pulsars. We summarize possible instrumental selection effects, showing that even if an unbiased sample of pulsars is still lacking, their influence on the results of the presented analysis is reduced by recent improvements in instrumentation and searching techniques.
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