Define the cross-ratio of four real or complex numbers as follows:
$$[a,b,c,d] = \frac{(a-c)(b-d)}{(a-d)(b-c)}.$$
Then its logarithm has the same symmetries as the curvature tensor:
$$\log[a,b,c,d] = -\log[b,a,c,d] = -\log[a,b,d,c] = \log[c,d,a,b].$$
Moreover, if $[a,b,c,d] = \lambda$, then $[b,c,a,d] = 1 - \lambda^{-1}$ and $[c,a,b,d] = (1-\lambda)^{-1}$, which implies the algebraic Bianchi identity:
$$\log[a,b,c,d] + \log[b,c,a,d] + \log[c,a,b,d] = -1.$$

Is there something behind these coincidences?