Context. The connection of cluster mergers with the presence of extended, diffuse radio sources in galaxy clusters is still debated. Aims: We aim to obtain new insights into the internal dynamics of the cluster Abell 2345. This cluster exhibits two non-symmetric radio relics well studied through recent, deep radio data. Methods: Our analysis is based on redshift data for 125 galaxies acquired at the Telescopio Nazionale Galileo and on new photometric data acquired at the Isaac Newton Telescope. We also use ROSAT/HRI archival X-ray data. We combine galaxy velocities and positions to select 98 cluster galaxies and analyze the internal dynamics of the cluster. Results: We estimate a mean redshift < z > = 0.1789 and a line-of-sight (LOS) velocity dispersion σV ~ 1070 km s-1. The two-dimensional galaxy distribution reveals the presence of three significant peaks within a region of ~1 h70-1 Mpc (the E, NW, and SW peaks). The spectroscopic catalog confirms the presence of these three clumps. The SW and NW clumps have similar mean velocities, while the E clump has a larger mean velocity (Δ Vrf ~ 800 km s-1); this structure causes the presence of the two peaks we find in the cluster velocity distribution. The difficulty in separating the galaxy clumps leads to a very uncertain mass estimate M ~ 2 × 1015 h70-1 M&sun;. Moreover, the E clump well coincides with the main mass peak as recovered from the weak gravitational lensing analysis and is off-set to the east from the BCG by ~1.3´. The ROSAT X-ray data also show a very complex structure, mainly elongated in the E-W direction, with two (likely three) peaks in the surface brightness distribution, which, however, are off-set from the position of the peaks in the galaxy density. The observed phenomenology agrees with the hypothesis that we are looking at a complex cluster merger occurring along two directions: a major merger along the ~E-W direction (having a component along the LOS) and a minor merger in the western cluster regions along the ~N-S direction, roughly parallel to the plane of the sky. The eastern radio relic is elongated in the direction perpendicular to that of the major merger, while the peculiar, western radio relic is elongated in the direction perpendicular to the bisecting of the two merger directions. Conclusions: Our scenario for the internal dynamics of Abell 2345 strongly supports the use of the “outgoing merger shocks” model to explain the two radio relics, suggesting a consistent justification for their asymmetry. Table 1 is only available in electronic form at http://www.aanda.org