A complete description of the multitudinous ways in which quantum particles can be entangled requires the use of high-dimensional abstract mathematical spaces. We report here a particularly interesting feature of the nine-dimensional convex set-endowed with Hilbert-Schmidt (Euclidean/flat) measure-composed of two-rebit (4 × 4) density matrices (ρ). To each ρ is assigned the product of its (nonnegative) determinant |ρ| and the determinant of its partial transpose |ρ P T |negative values of which, by the results of Peres and Horodecki, signify the entanglement of ρ.Integrating this product, |ρ||ρ P T | = |ρρ P T |, over the nine-dimensional space, using the indicated (HS) measure, we obtain the result zero. The two determinants, thus, form a pair of multivariate orthogonal polynomials with respect to HS measure. However, orthogonality does not hold, we find, if the symmetry of the nine-dimensional two-rebit scenario is broken slightly, nor with the use of non-flat measures, such as the prominent Bures (minimal monotone) measure-nor in the full HS extension to the 15-dimensional convex set of two-qubit density matrices. We discuss relationsinvolving the HS moments of |ρ P T |-to the long-standing problem of determining the probability that a generic pair of rebits/qubits is separable.