Crystal structure of double-perovskite type BaYMn2O5+δ was studied by high-temperature synchrotron X-ray diffraction (SXRD) under precisely controlled oxygen pressure to gain deeper understanding of the remarkable oxygen intake/release capability of this oxide. The in situ SXRD analysis at 750 °C revealed that this oxide undergoes a distinct structural change upon lowering oxygen pressure, from a slightly oxygen-deficient "δ = 1" phase (BaYMn2O5.89; P(O2) = 103 Pa) to an oxygen-vacancy ordered "δ = 0.5" phase (BaYMn2O5.51; P(O2) = 10 Pa). The BaYMn2O5.89 structure (orthorhombic Cmmm) involves statistical distribution of oxygen vacancies within the yttrium plane. Meanwhile, the BaYMn2O5.51 structure (orthorhombic Icma) contains arrays of pyramidal MnO5 and octahedral MnO6 forming an alternate ordering, which is stabilized by a particular Mn3+ orbital ordering with collective displacements of Y3+ arrays. Thus, the discontinuous change in the oxygen content can be attributed to the structural reconstruction with oxygen/vacancy redistribution accompanied by yttrium displacement organization.