Electron internal transport barriers (eITBs) are obtained in TCV with different heating and current drive schemes. They are sustained in steady-state conditions for several energy confinement and current redistribution times. In these scenarios, the density profile displays a different behaviour with respect to normal L-mode plasmas, with or without auxiliary heating. In fully non-inductive discharges developing an eITB, the density profile shape is strongly correlated with the electron temperature profile, with the normalized density gradient equal to 0.45 times the normalized temperature gradient, revealing the existence of a significant inward pinch of a thermodiffusive type. The coupling of the two profiles is observed from the 'foot' of the barrier inwards. The effect of small inductive current perturbations on fully non-inductive sustained eITBs shows that ∇n/n is only indirectly coupled to the current profile, through its effect on local confinement, contrary to the standard L-mode.