The gravity gradient tensor (whose components are the second derivatives of the gravitational potential) is a symmetric tensor that, ignoring the constraint imposed by Laplace's equation, contains only six independent components. When measured on a horizontal plane, these components generate, in the spectral domain, six power spectral densities (PSDs) and fifteen cross-spectra. The cross-spectra can be split into two groups: a real group and a pure imaginary group. If the source distribution is statistically stationary, 1D spectra can be found from the 2D spectra via the slice theorem. The PSDs form two power-sum rules that link all gradient components. The power-sum rules, in combination with further equalities between the power and cross-spectra, reduce the number of independent spectra to 13, a number reduced to seven if the power spectrum of the potential is assumed isotropic. The power-sum rules, cross-spectral phases, and coherence between components all provide information on the internal consistency of a set of gradiometry measurements. This information can be used to assess the noise, to determine the isotropy, and, for a self-similar source, to calculate the scaling factor and average depth. When applied to a data set collected in the North Sea, the power-sum rules reveal high-frequency noise that is distributed among only three of the gradient components; additionally, the coherences reveal the source to be anisotropic with a nonzero correlation length.