Summary: We consider the inverse elastic scattering problem of determining a three-dimensional diffraction grating profile from scattered waves measured above the structure. In general, a grating profile cannot be uniquely determined by a single incoming plane wave. We completely characterize and classify the bi-periodic polyhedral structures under the boundary conditions of the third and fourth kinds that cannot be uniquely recovered by only one incident plane wave. Thus we have global uniqueness for a polyhedral grating profile by one incident elastic plane wave if and only if the profile belongs to neither of the unidentifiable classes, which can be explicitly described depending on the incident field and the type of boundary conditions. Our approach is based on the reflection principle for the Navier equation and the reflectional and rotational invariance of the total field.