High-quality spectropolarimetry (range 417-860 nm; spectral resolution 1. 27 nm and 0.265 nm pixel-1) of the Type la supernova (SN Ia) 2001el was obtained with the ESO Very Large Telescope Melipal (+FORS1) at five epochs. The spectra a week before maximum and around maximum indicate photospheric expansion velocities of about 10,000 km s-1. Prior to optical maximum, the linear polarization of the continuum was ≈0.2%-0.3% with a constant position angle, showing that SN 2001el has a well-defined axis of symmetry. The polarization was nearly undetectable a week after optical maximum. The spectra are similar to those of the normally bright SN 1994D, with the exception of a strong double-troughed absorption feature seen around 800 nm (FWHM about 22 nm). The 800 nm feature is probably due to the Ca II IR triplet at very high velocities (20,000-26,000 km s-1) involving ∼0.004 M⊙ of calcium and perhaps 0.1 M⊙ total mass. The 800 nm feature is distinct in velocity space from the photospheric Ca II IR triplet and has a significantly higher degree of polarization (≈0.7%) and different polarization angle than the continuum. Taken together, these aspects suggest that this high-velocity calcium is a kinematically distinct feature with the matter distributed in a filament, torus, or array of "blobs" almost edge-on to the line of sight. This feature could thus be an important clue to the binary nature of SNe Ia, perhaps associated with an accretion disk, or to the nature of the thermonuclear burning, perhaps representing a stream of material ballistically ejected from the site of the deflagration to detonation transition. If modeled in terms of an oblate spheroid, the continuum polarization implies a minor to major axis ratio of around 0.9 if seen equator-on; this level of asymmetry would produce an absolute luminosity dispersion of about 0.1 mag when viewed at different viewing angles. If typical for SNe Ia, this would create an rms scatter of several hundredths of a magnitude around the mean brightness-decline relation. We discuss the possible implications of this scatter for the high-precision measurements required to determine the cosmological equation of state.