Re-based double perovskites (DPs) have garnered substantial attention due to their high Curie temperatures () and display of complex interplay of structural and metal-insulator transitions (MIT). Here we systematically study the ground-state electronic and structural properties for a family of Re-based DPs (=Sr, Ca and =Cr, Fe), which are related by a common low-energy Hamiltonian, using density functional theory + calculations. We show that the on-site interaction of Re induces orbital ordering (denoted C-OO), with each Re site having an occupied orbital and a C-type alternation among , resulting in an insulating state consistent with experimentally determined insulators , , and . The threshold value of for orbital ordering is reduced by inducing octahedral distortions of the same C-type wavelength (denoted C-OD), which serves as a structural signature of the orbital ordering; octahedral tilting also reduces the threshold. The C-OO and the concomitant C-OD are a spontaneously broken symmetry for the Sr-based materials (i.e., tilt pattern), while not for the Ca-based systems (i.e., tilt pattern). Spin-orbit coupling does not qualitatively change the physics of the C-OO/C-OD, but can induce relevant quantitative changes. We prove that a single set of capture the experimentally observed metallic state in and insulating states in other three systems. We predict that the C-OO is the origin of the insulating state in , and that the concomitant C-OD may be experimentally observed at sufficiently low temperatures (i.e., space group ) in pure samples. Additionally, given our prescribed values of , we show that the C-OO induced insulating state in will survive even if the C-OD amplitude is suppressed (e.g., due to thermal fluctuations). The role of the C-OO/C-OD in the discontinuous, temperature driven MIT in is discussed.