Rank-2 tensor fields of large-scale structure, e.g. a tensor field inferred from shapes of galaxies, open up a window to directly access 2 scalar, 2 vector and 2 tensor modes, where the scalar fields can be measured independently from the standard density field that is traced by distribution of galaxies.Here we develop an estimator of the multipole moments of coordinate-independent power spectra for the three-dimensional tensor field, taking into account the projection of the tensor field onto plane perpendicular to the line-of-sight direction. To do this, we find that a convenient representation of the power spectrum multipoles can be obtained by the use of the associated Legendre polynomials L m L (µ), instead of the standard one L (µ), in the form which allows for the fast Fourier transform estimations under the local plane-parallel (LPP) approximation. The formulation also allows us to obtain the Hankel transforms to connect the two-point statistics in Fourier and configuration space, which are needed to derive theoretical templates of the power spectrum including convolution of a survey window. To validate our estimators, we use the simulation data of the projected tidal field assuming a survey window that mimics the BOSS-like survey footprint. We show that the LPP estimators fairly well recover the multipole moments that are inferred from the global plane-parallel approximation. We find that the survey window causes a more significant change in the multipole moments of projected tensor power spectrum at k 0.1 hMpc −1 from the input power spectrum, than in the density power spectrum. Nevertheless, the theoretical predictions including the survey window effects match the window-convolved multipole moments measured from the simulations. The analysis method presented here paves the way for a cosmological analysis using three-dimensional tensor-type tracers of large-scale structure for current and future surveys.