Somewhat unexpectedly, the study of the family of twisted knots revealed a hidden structure behind exclusive Racah matrices $$\bar{S}$$, which control non-associativity of the representation product in a peculiar channel $$R\otimes \bar{R} \otimes R \longrightarrow R$$. These $$\bar{S}$$ are simultaneously symmetric and orthogonal and therefore admit two decompositions: as quadratic forms, $$\bar{S} \sim \mathcal{E}^{tr}\mathcal{E}$$, and as operators: $$\bar{T}\bar{S}\bar{T} = S T^{-1} S^{-1}$$. Here, $$\bar{T}$$ and T consist of the eigenvalues of the quantum $$\mathcal{R}$$-matrices in channels $$R\otimes \bar{R}$$ and $$R\otimes R$$, respectively, S is the second exclusive Racah matrix for $$\bar{R}\otimes R\otimes R \longrightarrow R$$ (still orthogonal, but no longer symmetric) and $$\mathcal{E}$$ is a triangular matrix. It can be further used to construct the KNTZ evolution matrix $$\mathcal{B}=\mathcal{E}\bar{T}^2\mathcal{E}^{-1}$$, which is also triangular and explicitly expressible through the skew Schur and Macdonald functions—what makes Racah matrices calculable. Moreover, $$\mathcal{B}$$ is somewhat similar to Ruijsenaars Hamiltonian, which is used to define Macdonald functions, and gets triangular in the Schur basis. Discovery of this pentad structure $$(\bar{T},\bar{S},S,\mathcal{E},\mathcal{B})$$, associated with the universal $$\mathcal{R}$$-matrix, can lead to further insights about representation theory, knot invariants and Macdonald–Kerov functions.