Abstract ABO₃ perovskites host a huge range of symmetry lowering structural distortions, each of which can tune, or even switch on or off, different functional properties due to the strong coupling between the lattice, spin and charge degrees of freedom in these materials. The sheer number of different meta-stable structures present in perovskites creates a challenge for materials design via theory and simulation. Here, we tackle this issue using a first principles structure searching method on a prototypical half-metallic perovskite, La_0.7Sr_0.3MnO₃, to predict how epitaxial strain can engineer structural and magnetic properties. We reveal a rich structural phase diagram through strain engineering in which the octahedral tilt pattern, and hence the crystal symmetry, is altered from the bulk. We show how the low-symmetry of the various phases in turn induces new structural modes, an increase in the magnetic anisotropy energy, and weak antiferromagnetic spin-canting.