The Belinski-Khalatnikov-Lifshitz (BKL) scenario (Belinskii et al., Adv Phys 19:525, 1970; Adv Phys 31:639, 1982; Ann Phys 70:301, 1971; Belinski, Int J Mod Phys D 23:1430016, 2014), which describes asymptotic dynamics of an anisotropic universe near the cosmological singularity, is found to manifest interesting behavior. Two basic variants are identified and analyzed in terms of the model’s diagonal Hamiltonian variables. The geometric picture of the dynamics may be seen as advancing of the system in the 3-dimensional velocity or momentum space. This motion takes place within a quadric of the “kinetic energy.” If the latter is an indefinite quadratic form, the quadric is a cone, as in the BKL case. Since the governing equations are time reversible, the description may be used for a collapse toward or expansion from the singularity. We find that the possible “subscenarios” of the collapse are: (1) unstable squeeze of the universe in all directions to a point; (2) stable approaching a limit in which the universe collapses in one or two directions, almost, but not completely, to a point, while stretching in the remaining two or one, like in Kasner’s models (Misner et al. Phys Rev D 87:084021, 2013). In the latter, the system reflects from a “potential wall,” which is another quadric, a two-sheet hyperboloid, and a new Kasner’s epoch begins. The scenario inevitably ends in the all-direction collapse; however, the approach is proved to be chaotic.