  The scale invariant Petrov classification of the Weyl tensor is linked to the
scale invariant combination of the Kasner index constraints, and the
Lifshitz-Khalatnikov Kasner index parametrization scheme turns out to be a
natural way of adapting to this symmetry, while hiding the permutation symmetry
that is instead made manifest by the Misner parametrization scheme. While not
so interesting for the Kasner spacetime by itself, it gives a geometrical
meaning to the famous Kasner map transitioning between Kasner epochs and Kasner
eras, equivalently bouncing between curvature walls, in the BLK-Mixmaster
dynamics exhibited by spatially homogeneous cosmologies approaching the initial
cosmological singularity and the inhomogeneous generalization of this dynamics.
