We perform an extensive study of Sr₃Cr₂O₇, the n = 2 member of the Ruddlesden-Popper Sr_n+1Cr_nO_3n+1 system. An antiferromagnetic ordering is clearly visible in the magnetization and the specific heat, which yields a huge transition entropy, R ln(6). By neutron diffraction as a function of temperature we have determined the antiferromagnetic structure that coincides with the one obtained from density functional theory calculations. It is accompanied by anomalous asymmetric distortions of the CrO₆ octahedra. Strong coupling and Lanczos calculations on a derived Kugel-Khomskii Hamiltonian yield a simultaneous orbital and moment ordering. Our results favor an exotic ordered phase of orbital singlets not originated by frustration.