We present exact results for steady-state density correlation functions in conserved-mass transport processes with it anisotropic, reflection-symmetric hopping on a d-dimensional hypercubic lattice. In addition to mass conservation, we consider center-of-mass (CoM) conservation, imposed either along a specific axis or along all axes. CoM-conserving dynamics is implemented through coordinated it multidirectional hopping of two equal chunks of masses in it opposite directions. While anisotropy and mass conservation are known to generate power-law density correlations C(bf x) ∼ 1/|bf x|ᵈ at large distance |bf x| ≫ 1 it [Phys. Rev. A bf 42, 1954 (1990)], an additional CoM conservation can qualitatively alter the nature of the power law. Indeed, when CoM is conserved in it all directions, the correlations decay faster - typically as C(bf x) ∼ 1/|bf x|⁽ᵈ⁺²⁾, regardless of the presence (or absence) of anisotropy. Consequently, the systems exhibit an extreme it hyperuniformity (``class I''), where the long-wavelength density fluctuations, despite the slow power-law decay, are anomalously suppressed. When CoM is conserved along particular (it not all) directions, the slower 1/|bf x|ᵈ power-law decay is recovered. The above behavior can be understood from an analogy between the correlation function and an electrostatic potential: While a (rank-2) quadrupolar charge distribution gives rise to the 1/|bf x|ᵈ power law, the 1/|bf x|⁽ᵈ⁺²⁾ power law originates from a higher-order (rank-4) multipolar charge distribution. These findings reveal a rich interplay between anisotropy and CoM conservation in nonequilibrium steady states.