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Any fourth rank plane tensor H obeying the "Hooke's" symmetries (H_ijkl = H_jikl = H_klij) can be split into three parts, behaving differently under the two-dimensional space rotation and belonging to the three different, mutually orthogonal, two-dimensional subspaces remaining stable under the rotation. Such representation leads to a convenient set of functionally independent invariants, vanishing of some of these invariants demarcating the transitions of the tensor to the higher symmetry class. A non-trivial effective condition of orthotropy has been obtained. Some problems concerning the necessary and complete set of measurements of the elastic properties are also encountered.