A reversible switching of the easy axis of magnetization for Ni on Cu(001) from in plane to out of plane is found by changing the partial pressure of hydrogen in the gas phase around the sample, allowing even for oscillations of the magnetization direction. A quantitative low-energy electron diffraction study of the diffracted intensity versus electron energy [I(E)] shows that the hydrogen-induced spin reorientation transition is accompanied by changes of the tetragonal distortion of the topmost Ni layer upon hydrogen adsorption. Surprisingly, the orientation switch to perpendicular to the surface comes with a relaxation, i.e., reduction of the film's tetragonal distortion rather than its amplification.