Context. We present the results of new spectral diagnostic investigations applied to high-resolution long-slit spectra obtained with the Hubble Space Telescope Imaging Spectrograph (HST/STIS) of the jet from the T Tauri star RW Aur. Aims. Our primary goal is to determine basic physical parameters (electron density n e and electron temperature T e , hydrogen ionisation fraction x e , total hydrogen density n H , radial velocity v r and the mass outflow rateṀ j ) along both the red-and blueshifted lobes of the RW Aur jet. Methods. The input dataset consists of seven long-slit spectra, of 0. 1 spatial resolution, taken with the STIS slit parallel to the jet, and stepped across it. We use the Bacciotti & Eislöffel (1999, A&A, 342, 717) Results. We were able to extract the parameters as far as 3. 9 in the red-and 2. 1 in the blueshifted beam. The electron density at the base of both lobes is close to the critical density for [S II] emission but then it decreases gradually with distance from the source. The range of electron temperatures derived for this jet (T e = 10 4 −2 × 10 4 K) is similar to those generally found in other outflows from young stars. The ionisation fraction x e varies between 0.04 and 0.4, increasing within the first few arcseconds and then decreasing in both lobes. The total hydrogen density, derived as n H = n e /x e , is on average 3.2 × 10 4 cm −3 and shows a gradual decrease along the beam. Variations of the above quantities along the jet lobes appear to be correlated with the position of knots. Combining the derived parameters with v r measured from the HST spectra and other characteristics available for this jet, we estimateṀ j following two different procedures. The mass-outflow rateṀ j is moderate and similar in the two lobes, despite the fact that the well-known asymmetry in the radial velocity persists close to the source. Using the results of the BE diagnostics we find averages along the first 2. 1 of both flows (a region presumably not yet affected by interaction with the jet environment) of 2.6 × 10 −9 M yr −1 for the red lobe and 2.0 × 10 −9 M yr −1 for the blueshifted flow, with an uncertainty of ±log M = 1.6. Conclusions. The fact that the derived mass outflow rate is similar in the two lobes appears to indicate that the central engine is constrained on the two sides of the system and that the observed asymmetries are due to environmental conditions. Possible suggestions for the origin of the differences are discussed. The RW Aur jet appears to be the second densest outflow from a T Tauri star studied so far, but its other properties are quite similar to those found in other jets from young stars, suggesting that a common acceleration mechanism operates in these sources.