The narrow NN-decoupled dibaryon resonance with a mass about 1956 MeV was reported in 2000 by DIB2γ Collaboration (JINR, Dubna) on the basis of the measurement of the two-photon energy spectrum in the reaction pp → pp2γ at 216 MeV. The most probable quantum numbers J P = 1 + prevent the resonance from decaying into two protons due to the exclusion principle, while the pionic decays are impossible energetically. The signiˇcance of this resonance (called d * 1 (1956)) for the interpretation of a few other exclusive and inclusive reactions connected with the photon(s) production in nucleon collisions with nucleons and nuclei at different energies is discussed. The importance is stressed of ongoing and planned studies of the elastic and inelastic Compton scattering on the lightest nuclei for collecting information on the structure and dynamics of d * 1 (1956) which can shed light on its nature.