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We investigate by first‐principles the structural behavior of CaSiO₃ perovskite up to lower mantle pressures. We confirm that the cubic perovskite modification is unstable at all pressures. The zero Kelvin structure is stabilized by SiO₆ octahedral rotations that lower the symmetry to tetragonal, orthorhombic, rhombohedral, or to a cubic supercell. The resulting structures have comparable energies and equation of state parameters. This suggests that relatively small deviatoric/shear stresses might induce phase transformations between these various structures softening some elastic moduli, primarily the shear modulus. The seismic signature accompanying a local increase in CaSiO₃ content should be a positive density anomaly and a negative V_S anomaly.