  The $\alpha$-$T_3$ system undergoes a topological phase transition(TPT)
between two distinct quantum spin-Hall phases across $\alpha=0.5$ when the
spin-orbit interaction of Kane-Mele type is taken into consideration. As a
hallmark of such a TPT, we find that the Berry curvature and the orbital
magnetic moment change their respective signs across the TPT. We also find the
trails of the TPT in another physical observable, namely, the orbital
magnetization(OM) that can be, in principle, detected experimentally through
the circular dichroism associated with optical absorption. The topological
features of the OM are understood in terms of valley and spin physics. The
valley-resolved OM(VROM) and the spin-resolved OM(SROM) exhibit interesting
characteristics related to the valley and the spin Chern number when the
chemical potential is tuned in the forbidden gap(s) of the energy spectrum. In
particular, we find that the slope of the VROM versus the chemical potential in
the forbidden gap changes its sign abruptly across the TPT, which is also
consistent with the corresponding change in the valley Chern number. Moreover,
the slope of the SROM demonstrates a sudden jump by one unit of $e/h$ (where
$e$ is the electronic charge and $h$ is the Planck's constant) across the TPT,
which is also in agreement with the corresponding change in the spin Chern
number. It is further seen that a definite spin-valley optical selection rule
governs the circular dichroism. The $k$-resolved degree of the optical
polarization and the low-frequency differential optical absorbance manifest
sign change across the TPT.
