Large changes in the magnetization of ferromagnetic films can be electrically driven by non-180° ferroelectric domain switching in underlying substrates, but the shear components of the strains that mediate these magnetoelectric effects have not been considered so far. Here we reveal the presence of these shear strains in a polycrystalline film of Ni on a 0.68Pb(Mg1/3Nb2/3)O3-0.32PbTiO3 substrate in the pseudo-cubic (011)pc orientation. Although vibrating sample magnetometry records giant magnetoelectric effects that are consistent with the hitherto expected 90° rotations of a global magnetic easy axis, high-resolution vector maps of magnetization (constructed from photoemission electron microscopy data, with contrast from X-ray magnetic circular dichroism) reveal that the local magnetization typically rotates through smaller angles of 62-84°. This shortfall with respect to 90° is a consequence of the shear strain associated with ferroelectric domain switching. The non-orthogonality represents both a challenge and an opportunity for the development and miniaturization of magnetoelectric devices.