The main features of this paper are the utilization of inherent two-dimensional symmetries and the development of accurate angular quadrature coordinates and weights especially suited for the net and/or partial currents and all the net and/or partial moments of the neutron flux up to a given order. Three classes of true analogs of the one-dimensional single-Gauss and double-Gauss are considered for two-dimensional x-y problems with rectangular spatial mesh subdivisions. The first is single-range quadrature, most suitable for the asymptotic regions where the vector flux of neutrons can be well approximated by polynomials in Ωx and Ωy defined over the entire unit sphere of angular directions Ω. This quadrature can be used whenever distances between material interfaces are large with respect to the neutron mean-free-path (mfp). The second is double-range quadrature, most suitable at material interfaces where the unit sphere can be split into two hemispheres, one in each material region, and the vector flux can be well approximated by two possibly distinct polynomials in Ωx and Ωy, one in each hemisphere. This quadrature can be used whenever material interfaces and currents are important along either the x or the y direction but not both. The third is quadruple-range quadrature, most suitable at corners where the unit sphere can be split into four quadrants and the vector flux can be well approximated by four possibly distinct polynomials in Ωx and Ωy, one in each quadrant. This quadrature explicitly allows for discontinuities at corners and is appropriate for highly heterogeneous problems where distances between material corners are small with respect to the mfp. For simplicity, only product formulas are considered, where the angular integrals are split into separate integrals over polar and azimuthal directions. The quadratures developed are initially based on the standard choice of the polar angle θ with respect to the z axis and the azimuthal angle with respect to the x axis in the x-y plane. Quadratures based on 90-deg rotations of the angular coordinate representation are also considered where, for example, θ is measured with respect to the y axis and with respect to the x axis or vice versa. Proper guidelines for the choice of polar angle coordinates, given a certain choice of azimuthal angle coordinates, or vice versa, are established by the compatibility requirement. Compatibility of a product formula requires integrals over the polar angle and over the azimuthal angle to have the same order of accuracy. Numerical comparisons of the various quadratures for two standard problems are summarized and suggest that the new quadruple-range quadrature and the double-range rotated Chebyshev-Gauss double-Gauss quadrature can prove very competitive for several transport problems in both reducing ray effects and improving accuracy.