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The spin 2 representations of the rotation group contains
eigenvectors of one of the generators (call it L_3) with eignvalues
M=_2,-1,0,1,2 (as you correctly wrote in your posting). A rotation
of \phi around the 3-axis induces a change exp(iM\phi) in each respective
eigenvector. Accordingly, the eigenvectors corresponding to M=+-1 will
change sign when subjected to a rotation of \pi.
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A graviton eigenvector, therefore, is invariant under rotations
of \pi around the axis along its direction of motion. We do not consider
rotations about other axes because they would change the direction of
motion.