The high-pressure sequence of the perovskite polytypes of BalrO(3) has been investigated in the pressure range up to 10 GPa. At ambient pressure the so-called "9R" polytype has been prepared by solid-state reaction and slow cooling in air to yield an almost fully oxygen-stoichiometric BalrO(296(1)) composition. The crystal structure has been refined from XRD data in the monoclinic C2/m space group with a = 10.0046(3) angstrom, b = 5.75362(14) angstrom, c = 15.1839(4) angstrom, beta = 103.27(1)degrees; it contains trimers of facesharing octahedra (or Ir3O12 trioctahedra) that are linked by their vertices to form columns parallel to the c-axis with a stacking of layers of corner sharing (c) and face sharing (h) IrO6 octahedra along the sequence hhchhc. This structure is stable up to 3 GPa; at 4 GPa a new 5H polytype has been stabilized as a pure phase. The crystal structure has been solved by ab initio procedures from powder XRD data. It is monoclinic with a = 9.9511(2)angstrom, b = 5.7503(1) angstrom, c = 13.71003(3)angstrom, beta = 118.404(2)degrees, and it was refined in the C2/m space group from NPD data collected at room temperature. This polytype can be described as a stacking of IrO6 octahedra along the sequence hchcc. The structure contains chains of double dimer units of facesharing octahedra; the twin dimers are connected to single layers of vertex-sharing octahedra, forming infinite chains along c. This is a unique stacking that, with this repetition length, has never been described before among the hexagonal polytypes of ABO(3) perovskites. The 5H polytype is stable in a narrow pressure range; at 5 GPa the 6H structure is formed, stable up to 10 GPa. The 6H-BaIrO3 polytype is monoclinic, space group C2/c, with a = 5.7483(2)angstrom, b = 9.9390(3)angstrom, c = 14.3582(5)angstrom, beta = 91.319(2)degrees. The structure consists of dimers of face-sharing octahedra separated by single corner-sharing octahedra, showing the sequence hcchcc along the c-axis. At 10 GPa the cubic 3C perovskite structure could be identified as a minority phase, with a = 4.0611(7) angstrom, defined in the Pm (3) over barm space group. The precarious stability of the 5H polytype, as well as the novel pressure sequence displayed by BalrO(3) that is distinct from the classical sequence 9R-4H-6H-3C exhibited by many transition metal oxides, for instance BaRuO3, is a result of the particular stability of the "9R' ambient-pressure structure, which is reinforced by a strong Ir-Ir bond across the octahedral faces, and the Ir-Ir Coulombic repulsion across shared faces that destabilizes the 4H polytype relative to the 6H phase to allow stabilization of the hybrid 5H polytype in a narrow presure range.