Context. The α Centauri binary system, owing to its duplicity, proximity and brightness, and its components' likeness to the Sun, is a fundamental calibrating object for the theory of stellar structure and evolution and the determination of stellar atmospheric parameters. This role, however, is hindered by a considerable disagreement in the published analyses of its atmospheric parameters and abundances. Aims. We report a new spectroscopic analysis of both components of the α Centauri system, compare published analyses of the system, and attempt to quantify the discrepancies still extant in the determinations of the atmospheric parameters and abundances of these stars. Methods. The analysis is differential with respect to the Sun, based on spectra with R = 35 000 and signal-to-noise ratio ≥1000, and employed spectroscopic and photometric methods to obtain as many independent T eff determinations as possible. We also check the atmospheric parameters for consistency against the results of the dynamical analysis and the positions of the components in a theoretical HR diagram. Results. The spectroscopic atmospheric parameters of the system are found to be T eff = (5847 ± 27) K, [Fe/H] = +0.24 ± 0.03, log g = 4.34 ± 0.12, and ξ t = 1.46 ± 0.03 km s −1 , for α Cen A, and T eff = (5316 ± 28) K, [Fe/H] = +0.25 ± 0.04, log g = 4.44 ± 0.15, and ξ t = 1.28 ± 0.15 km s −1 for α Cen B. The parameters were derived from the simultaneous excitation & ionization equilibria of Fe I and Fe II lines. T eff s were also obtained by fitting theoretical profiles to the Hα line and from photometric calibrations. Conclusions. We reached good agreement between the three criteria for α Cen A. For α Cen B the spectroscopic T eff is ∼140 K higher than the other two determinations. We discuss possible origins of this inconsistency, concluding that the presence of non-local thermodynamic equilibrium effects is a probable candidate, but we note that there is as yet no consensus on the existence and cause of an offset between the spectroscopic and photometric T eff scales of cool dwarfs. The spectroscopic surface gravities also agree with those derived from directly measured masses and radii. An average of three independent T eff criteria leads to T eff (A) = (5824 ± 26) K and T eff (B) = (5223 ± 62) K. The abundances of Na, Mg, Si, Mn, Co, and Ni and, possibly, Cu are significantly enriched in the system, which also seems to be deficient in Y and Ba. This abundance pattern can be deemed normal in the context of recent data on metal-rich stars. The position of α Cen A in an up-to-date theoretical evolutionary diagram yields a good match of the evolutionary mass and age (in the 4.5 to 5.3 Gyr range) with those from the dynamical solution and seismology, but only marginal agreement for α Cen B, taking into account its more uncertain T eff .