Context: The mechanisms giving rise to diffuse radio emission in galaxy clusters, and in particular their connection with cluster mergers, are still debated. Aims: We aim to obtain new insights into the internal dynamics of the cluster Abell 1240, which appears to contain two roughly symmetric radio relics, separated by ~2 h_70-1 Mpc. Methods: Our analysis is based mainly on redshift data for 145 galaxies mostly acquired at the Telescopio Nazionale Galileo and on new photometric data acquired at the Isaac Newton Telescope. We also use X-ray data from the Chandra archive and photometric data from the Sloan Digital Sky Survey (Data Release 7). We combine galaxy velocities and positions to select 89 cluster galaxies and analyze the internal dynamics of the Abell 1237 + Abell 1240 cluster complex, Abell 1237 being a close companion of Abell 1240 in its southern direction. Results: We estimate similar redshifts for Abell 1237 and Abell 1240, < z > = 0.1935 and < z > = 0.1948, respectively. For Abell 1237, we estimate a line-of-sight (LOS) velocity dispersion of σV ~ 740 km s-1and a mass of M ~ 6 × 1014 h_70-1 M&sun;. For Abell 1240, we estimate a LOS σV ~ 870 km s-1and a mass in the range M ~ 0.9-1.9 × 1015 h_70-1 M&sun;, which takes account of its complex dynamics. Abell 1240 is shown to have a bimodal structure with two galaxy clumps roughly aligned along its N-S direction, the same as defined by the elongation of its X-ray surface brightness and the axis of symmetry of the relics. The two brightest galaxies of Abell 1240, associated with the northern and southern clumps, are separated by a LOS rest-frame velocity difference Vrf ~ 400 km s-1and a projected distance D ~ 1.2 h_70-1 Mpc. The two-body model agrees with the hypothesis that we are looking at a cluster merger that occurred largely in the plane of the sky, the two galaxy clumps being separated by a rest-frame velocity difference Vrf ~ 2000 km s-1at a time of 0.3 Gyr after the crossing core, while Abell 1237 is still infalling onto Abell 1240. Chandra archive data confirm the complex structure of Abell 1240 and allow us to estimate a global X-ray temperature of TX = 6.0±0.5 keV. Conclusions: In agreement with the findings from radio data, our results for Abell 1240 strongly support the “outgoing merger shocks” model to explain the presence of the relics.