We determine the elasticity of FeSiO3 perovskite for various spin configurations using density-functional theory calculations. The elastic moduli and the bulk seismic wave velocities are weakly affected by the spin transition. However we show that the intrinsic differences in seismic anisotropy between the high-spin and low-spin phases of Fe-bearing perovskite coupled with lattice preferred orientation that can develop due and during the convection may lead to distinct seismic signatures between the top and the bottom of the lower mantle. These signatures should be detectable in observations and they need to be taken into account in tomographic studies of the Earth's lower mantle.