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 VS anomaly.