We study a sample of 16 Type Ia supernovae (SNe Ia) having both spectroscopic and photometric observations within 2-3 days after the first light. The early B - V colors of such a sample tend to show a continuous distribution. For objects with normal ejecta velocity (NV), the C ii lambda 6580 feature is always visible in the early spectra, while it is absent or very weak in the high-velocity (HV) counterpart. Moreover, the velocities of the detached high-velocity features (HVFs) of the Ca II near-IR triplet (CaIR3) above the photosphere are found to be much higher in HV objects than in NV objects, with typical values exceeding 30,000 km s(-1) at 2-3 days. We further analyze the relation between the velocity shift of late-time [Fe II] lines (v([Fe II])) and host galaxy mass. We find that all HV objects have redshifted v([Fe II]), while NV objects have both blue- and redshifted v([Fe II]). It is interesting to point out that the objects with redshifted v([Fe II]) are all located in massive galaxies, implying that HV and a portion of NV objects may have similar progenitor metallicities and explosion mechanisms. We propose that, with a geometric/projected effect, the He-detonation model may account for the similarity in birthplace environment and the differences seen in some SNe Ia, including B - V colors, C II features, CaIR3 HVFs at early times, and v([Fe II]) in the nebular phase. Nevertheless, some features predicted by He-detonation simulation, such as the rapidly decreasing light curve, deviate from the observations, and some NV objects with blueshifted nebular v([Fe II]) may involve other explosion mechanisms.