Aims: XO-2 is the first confirmed wide stellar binary system where the almost twin components XO-2N and XO-2S have planets, and it is a peculiar laboratory in which to investigate the diversity of planetary systems. This stimulated a detailed characterization study of the stellar and planetary components based on new observations. Methods: We collected high-resolution spectra with the HARPS-N spectrograph and multi-band light curves. Spectral analysis led to an accurate determination of the stellar atmospheric parameters and characterization of the stellar activity, and high-precision radial velocities of XO-2N were measured. We collected 14 transit light curves of XO-2Nb used to improve the transit parameters. Photometry provided accurate magnitude differences between the stars and a measure of their rotation periods. Results: The iron abundance of XO-2N was found to be +0.054 dex greater, within more than 3σ, than that of XO-2S. The existence of a long-term variation in the radial velocities of XO-2N is confirmed, and we detected a turnover with respect to previous measurements. We suggest the presence of a second massive companion in an outer orbit or the stellar activity cycle as possible causes of the observed acceleration. The latter explanation seems more plausible with the present dataset. We obtained an accurate value of the projected spin-orbit angle for the XO-2N system (λ = 7° ± 11°), and estimated the real 3D spin-orbit angle (ψ =27+12-27 degrees). We measured the XO-2 rotation periods, and found a value of P = 41.6 ± 1.1 days in the case of XO-2N, in excellent agreement with the predictions. The period of XO-2S appears shorter, with an ambiguity between 26 and 34.5 days that we cannot solve with the present dataset alone. The analysis of the stellar activity shows that XO-2N appears to be more active than the companion, perhaps because we sampled different phases of their activity cycle, or because of an interaction between XO-2N and its hot Jupiter that we could not confirm. Based on observations made (i) with the Italian Telescopio Nazionale Galileo (TNG), operated on the island of La Palma by the INAF - Fundacion Galileo Galilei (Spanish Observatory of Roque de los Muchachos of the IAC); (ii) with the Copernico and Schmidt telescopes (INAF - Osservatorio Astrofisico di Padova, Asiago, Italy); (iii) with the IAC-80 telescope at the Teide Observatory (Instituto de Astrofísica de Canarias, IAC); (iv) at the Serra la Nave "M.G. Fracastoro" Astronomical Observatory (INAF - Osservatorio Astrofisico di Catania); (v) at the Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA).RV data (including Table 7) and photometry data are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/575/A111