(Abridged) We present a spectral analysis of a new, flux-limited sample of 72 X-ray selected clusters of galaxies identified with the X-ray Telescope (XRT) on board the Swift satellite down to a flux limit of ~10-14 erg/s/cm2 (SWXCS, Tundo et al. 2012). We carry out a detailed X-ray spectral analysis with the twofold aim of measuring redshifts and characterizing the properties of the Intra-Cluster Medium (ICM). Optical counterparts and spectroscopic or photometric redshifts are obtained with a cross-correlation with NED. Additional photometric redshifts are computed with a dedicated follow-up program with the TNG and a cross-correlation with the SDSS. We also detect the iron emission lines in 35% of the sample, and hence obtain a robust measure of the X-ray redshift zX. We use zX whenever the optical redshift is not available. Finally, for all the sources with measured redshift, background-subtracted spectra are fitted with a mekal model. We perform extensive spectral simulations to derive an empirical formula to account for fitting bias. The bias-corrected values are then used to investigate the scaling properties of the X-ray observables. Overall, we are able to characterize the ICM of 46 sources. The sample is mostly constituted by clusters with temperatures between 3 and 10 keV, plus 14 low-mass clusters and groups with temperatures below 3 keV. The redshift distribution peaks around z~0.25 and extends up to z~1, with 60% of the sample at 0.1<z<0.4. We derive the Luminosity-Temperature relation for these 46 sources, finding good agreement with previous studies. The quality of the SWXCS sample is comparable to other samples available in the literature and obtained with much larger X-ray telescopes. Our results have interesting implications for the design of future X-ray survey telescopes, characterised by good-quality PSF over the entire field of view and low background.
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