Aims. We study the optical spectroscopic properties of Type Ia Supernova (SN Ia) 2004dt, focusing our attention on the early epochs. Methods. Observation triggered soon after the SN 2004dt discovery allowed us to obtain a spectrophotometric coverage from day -10 to almost one year (~353 days) after the B band maximum. Observations carried out on an almost daily basis allowed us a good sampling of the fast spectroscopic evolution of SN 2004dt in the early stages. To obtain this result, low-resolution, long-slit spectroscopy was obtained using a number of facilities. Results. This supernova, which in some absorption lines of its early spectra showed the highest degree of polarization ever measured in any SN Ia, has a complex velocity structure in the outer layers of its ejecta. Unburnt oxygen is present, moving at velocities as high as ~16,700 km/s, with some intermediate-mass elements (Mg, Si, Ca) moving equally fast. Modeling of the spectra based on standard density profiles of the ejecta fails to reproduce the observed features, whereas enhancing the density of outer layers significantly improves the fit. Our analysis indicates the presence of clumps of high-velocity, intermediate-mass elements in the outermost layers, which is also suggested by the spectropolarimetric data. ; Comment: 13 pages, 15 figures, accepted for pubblication in Astronomy and Astrophysics