The X-Ray Telescope (XRT) on board Swift was mainly designed to provide detailed position, timing and spectroscopic information on Gamma-Ray Burst (GRB) afterglows. During the mission lifetime the fraction of observing time allocated to other types of source has been steadily increased. In this paper, we report on the results of the in-flight calibration of the timing capabilities of the XRT in Windowed Timing read-out mode. We use observations of the Crab pulsar to evaluate the accuracy of the pulse period determination by comparing the values obtained by the XRT timing analysis with the values derived from radio monitoring. We also check the absolute time reconstruction measuring the phase position of the main peak in the Crab profile and comparing it both with the value reported in literature and with the result that we obtain from a simultaneous Rossi X-Ray Timing Explorer (RXTE) observation. We find that the accuracy in period determination for the Crab pulsar is of the order of a few picoseconds for the observation with the largest data time span. The absolute time reconstruction, measured using the position of the Crab main peak, shows that the main peak anticipates the phase of the position reported in literature for RXTE by ~270 microseconds on average (~150 microseconds when data are reduced with the attitude file corrected with the UVOT data). The analysis of the simultaneous Swift-XRT and RXTE Proportional Counter Array (PCA) observations confirms that the XRT Crab profile leads the PCA profile by ~200 microseconds. The analysis of XRT Photodiode mode data and BAT event data shows a main peak position in good agreement with the RXTE, suggesting the discrepancy observed in XRT data in Windowed Timing mode is likely due to a systematic offset in the time assignment for this XRT read out mode.
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