The direct search for dark matter in the form of weakly interacting massive particles (WIMP) is performed by detecting nuclear recoils (NR) produced in a target material from the WIMP elastic scattering. A promising experimental strategy for direct dark matter search employs argon dual-phase time projection chambers (TPC). One of the advantages of the TPC is the capability to detect both the scintillation and charge signals produced by NRs. Furthermore, the existence of a drift electric field in the TPC breaks the rotational symmetry: the angle between the drift field and the momentum of the recoiling nucleus can potentially affect the charge recombination probability in liquid argon and then the relative balance between the two signal channels. This fact could make the detector sensitive to the directionality of the WIMP-induced signal, enabling unmistakable annual and daily modulation signatures for future searches aiming for discovery. The Recoil Directionality (ReD) experiment was designed to probe for such directional sensitivity. The TPC of ReD was irradiated with neutrons at the INFN Laboratori Nazionali del Sud, and data were taken with 72 keV NRs of known recoil directions. The direction-dependent liquid argon charge recombination model by Cataudella et al. was adopted and a likelihood statistical analysis was performed, which gave no indications of significant dependence of the detector response to the recoil direction. The aspect ratio R of the initial ionization cloud is estimated to be 1.037 +/- 0.027 and the upper limit is R < 1.072 with 90% confidence level