The thermal expansion of a low symmetry crystal can be much more interesting than the lattice parameter expansion would suggest. Here, the complete thermal expansion tensors for monoclinic and tetragonal phases of ZrO2 and HfO2 have been measured in air, by high-resolution, high-temperature X-ray diffraction. These results reveal the highly anisotropic nature of thermal expansion in the monoclinic phase as well as a cooperative movement of ions and the existence of a zero thermal expansion plane.