The use of piecewise constant functions (PCFs) in two-angle linear transport theory to represent the scattering cross sections σ(v), v ∈ [-1,1], and the angular scattering source density S(), ≡ (μ, φ) ∈ on a partition (SN or finite element discretization, for example) of the unit sphere of directions is considered. Average oriented transition cross sections σtn (±,B',B) describe scattering from ≡ (, )∈ B’ ⊂  to ≡ (μ,φ)∈ B ⊂ with the constraint 0< ±(φ - φ') <π. Unit steps σ(v) = H(v —γ) and σ(v) = δ(v — γ) are pretreated on an “intrinsic” γ grid for the chosen partition. All σtn(±,B',B) are derived by interpolation. The invariance properties of the σtn’s and the permitted B'→B transition (σtn > 0) are identified. Then, the PCF representation of S() is obtained with a minimum of work. Angular rebalancing restores the correct zeroth- and first-order angular moments without losing the nonnegativity of σtn and S. The preferential domains of application of this PCF method and the classical spherical harmonics method (which may violate nonnegativity) are discussed.