The cubic-to-orthorhombic structural transition occurring in CsH at a pressure of about 17 GPa is studied by ab initio calculations. The relative stability of the competing structures and the transition pressure are correctly predicted. We show that this pressure-induced first-order transition is intimately related to a displacive second-order transition which would occur upon application of a shear strain to the (110) planes. The resulting instability is rationalized in terms of the pressure-induced modifications of the electronic structure.