The anelastic spectrum ͑dynamic Young's modulus and elastic energy absorption͒ of La 2 CuO 4ϩ␦ has been measured between 1 and 700 K with 0Ͻ␦Ͻ0.02. The spectrum of stoichiometric La 2 CuO 4 in the lowtemperature orthorhombic ͑LTO͒ phase is dominated by two intense relaxation processes which cause softenings of 16% around 150 K and 9% below 30 K at f ϳ1 kHz. The relaxation at 150 K is attributed to the presence of a fraction of the CuO 6 octahedra which are able to change their tilted configuration by thermal activation between orientations which are nearly energetically equivalent, possibly within the twin boundaries. The relaxation below 30 K is governed by tunneling, and involves a considerable fraction of the lattice atoms. It is proposed that the double-well potentials for the low-temperature relaxation are created by the tendency of the LTO phase to form low-temperature tetragonal ͑LTT͒ domains, which however are not stabilized like when La is partially substituted with Ba. On doping with excess O, the relaxation rates of these processes are initially enhanced by hole doping, while their intensities are depressed by lattice disorder; an explanation of this behavior is provided. Excess O also causes two additional relaxation processes. The one appearing at lower values of ␦ is attributed to the hopping of single interstitial O 2Ϫ ions, with a hopping rate equal to Ϫ1 ϭ2ϫ10Ϫ14 exp(Ϫ5600/T) s. The second process is slower and can be due to O pairs or other complexes containing excess O. ͓S0163-1829͑98͒03210-X͔