This paper presents the analysis of high-resolution Stokes observations of eight different umbral dots in a sunspot. The spectra were recorded with the La Palma Stokes Polarimeter, attached to the Swedish Vacuum Solar Telescope. The observed line profiles have been inverted to yield the height stratifications of temperature, magnetic field, and line-of-sight velocity, as well as their respective Wilson depressions. We report on systematic differences in the properties of umbral dots with respect to the nearby umbra, including small upflows (~100 m s-1), higher temperatures (~1 kK), and weaker fields (~500 G) with more horizontal orientations (~10°). The field weakening is strongly correlated with the Wilson depression, suggesting that it may be due to an opacity effect (as one is looking at higher layers). The inclination excess, on the other hand, is real and cannot be ascribed to formation height issues. The results obtained from our semiempirical modeling are discussed within the context of the currently existing scenarios for the subsurface structure of sunspots. The observational signatures revealed by our analysis fit well within both the ``spaghetti'' and the monolithic models.