We study rotational dynamics induced by the recoil effect in diatomic molecules using time-resolved two color X-ray pump-probe spectroscopy. A short pump X-ray pulse ionizes a valence electron inducing molecular rotational wave packet, while the second time delayed X-ray pulse probes the dynamics. An accurate theoretical description is used for analytical discussions and numerical simulations. Our main attention is paid to the two interference effects that influence the recoil-induced dynamics: i) Cohen-Fano (CF) two-center interference between partial ionization channels in diatomics; ii) interference between the recoil-excited rotational levels manifesting as the rotational revival structures in the time-dependent absorption of the probe pulse. The time-dependent X-ray absorption is computed for the heteronuclear CO and homonuclear N$_2$ molecules as showcases. It is found that the effect of CF interference is comparable with the contribution from independent partial ionization channels, especially for the low photoelectron kinetic energy case. The amplitude of the recoil-induced revival structures for the individual ionization decreases monotonously with decrease of the photoelectron energy, while the amplitude of the CF contribution remains sufficient even at the photoelectron kinetic energy below 1 eV. The profile and intensity of the CF interference depend on the phase difference between the individual ionization channels related to the parity of the molecular orbital emitting the photoelectron. This phenomenon provides a sensitive tool for symmetry analysis of the molecular orbitals.