Stellar jets can be highly asymmetric and have multiple velocity components. To clarify the origin of jet asymmetries and constrain their launch mechanism we study the physical and kinematical structure of the flow emitted by DG Tau B. The analysis of deep spectra taken at the KECK telescope allows us to infer the physical properties (the electron and total density, ne and nh, the ionisation fraction, xe, and the temperature, te) and the spatial distribution of the velocity components in the two jet lobes. The presence of dust grains in the jet is investigated by estimating the gas-phase abundance of calcium with respect to its solar value. At the base of the jet the lines are broad (~100 km/s) and up to three velocity components are detected. At 5" from the source, however, only the denser and more excited high velocity components survive and the lines are narrower (~10-30 km/s). The jet is strongly asymmetric both in velocity and in its physical structure. The red lobe, slower (~140 km/s) and more collimated, presents low ionisation fractions (xe~0.1-0.4) and temperatures (te<5e3 K), while the total density is up to ~2.5e4 ccm. The blue lobe, faster (~-320 km/s) and less collimated, is also less dense (nh~1e4 ccm) but highly excited (te up to ~5e4 K and xe up to 0.9). The estimated mass loss rate is similar in the two lobes (~6-8e-9 Msol/yr), suggesting that the ejection power is comparable on the two sides of the system, as expected from a magneto-centrifugal ejection mechanism, and that the observed asymmetries are due to different mass load and propagation properties in an inhomogeneous environment. Calcium is strongly depleted, indicating that the jet contains dust grains and, therefore, should originate from a region of the disk extending beyond the dust sublimation radius. The depletion is lower for higher velocities, consistent with dust destruction by shocks.