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This paper describes how phase-sensitive rotation invariants for three-dimensional data may be obtained. A “bispectrum” is formulated for rotations, and its properties are derived for spherical harmonic coefficients as well as for moments. The bispectral invariants offer improved discrimination over previously published magnitude-only invariants. They are able to distinguish rotations from reflections, as well as rotations of an entire shape from component-wise rotations of elements of the shape. As experiments show, they provide robust performance for both surface and voxel data.