  In this manuscript we identify the main differences between the effects of
Kramers symmetry on the systems with even and odd number of electrons, the ways
how the aforementioned symmetry affects the structure of the Conical Seams
(CSs), and how it shows up in semiclassical propagation of nuclear wavepackets,
crossing the CSs. We identify the topological invariants, associated with CSs,
in three cases: even and odd number of electrons with time-reversal symmetry,
as well as absence of the latter. We obtain asymptotically exact semiclassical
analytical solutions for wavepackets scattered on a CS for all three cases,
identify topological features in a non-trivial shape of the scattered
wavepacket, and connect them to the topological invariants, associated with
CSs. We argue that, due to robustness of topology, the non-trivial wavepacket
structure is a topologically protected evidence of a wavepacket having passed
through a CS, rather than a feature of a semiclassical approximation.
