Spectropolarimetry enables us to measure the geometry and chemical structure of the ejecta in supernova explosions, which is fundamental for the understanding of their explosion mechanism(s) and progenitor systems. We collected archival data of 35 Type Ia supernovae (SNe Ia), observed with Focal Reducer and Low-Dispersion Spectrograph (FORS) on the Very Large Telescope at 127 epochs in total. We examined the polarization of the Si II lambda 6355 angstrom line (p(Si II)) as a function of time, which is seen to peak at a range of various polarization degrees and epochs relative to maximum brightness. We reproduced the Delta m(15)-p(Si II) relationship identified in a previous study, and show that subluminous and transitional objects display polarization values below the Delta m(15)-p(Si II) relationship for normal SNe Ia. We found a statistically significant linear relationship between the polarization of the Si II lambda 6355 angstrom line before maximum brightness and the Si II line velocity and suggest that this, along with the Delta m(15)-p(Si II) relationship, may be explained in the context of a delayed-detonation model. In contrast, we compared our observations to numerical predictions in the Delta m(15)-v(Si II) plane and found a dichotomy in the polarization properties between Chandrasekhar and sub-Chandrasekhar mass explosions, which supports the possibility of two distinct explosion mechanisms. A subsample of SNe displays evolution of loops in the q-u plane that suggests a more complex Si structure with depth. This insight, which could not be gleaned from total flux spectra, presents a new constraint on explosion models. Finally, we compared our statistical sample of the Si II polarization to quantitative predictions of the polarization levels for the double-detonation, delayed-detonation, and violent-merger models.