  Since violations of inequalities implied by non-contextual and local hidden
variable theories are observed, it is essential to determine the set of
(non-)contextual states. Along this direction, one should determine the
conditions under which quantum contextuality is observed. It is also important
to determine how one can find maximally contextual qutrits. In this work, we
revisit the Klyachko-Can-Binicio\u{g}lu-Shumovsky (KCBS) scenario where we
observe a five-measurement state-dependent contextuality. We investigate
possible symmetries of the KCBS pentagram, i.e., the conservation of the
contextual characteristic of a qutrit-system. For this purpose, the KCBS
operator including five cyclic measurements is rotated around the Z-axis. We
then check a set of rotation angles to determine the contextuality and
non-contextuality regions for the eigenstates of the spin-1 operator for an
arbitrary rotation. We perform the same operation for the homogeneous linear
combination of the eigenstates with spin values +1 and -1. More generally, we
work on the real subgroup of the three dimensional Hilbert space to determine
the set of (non-)contextual states under certain rotations in the physical
Euclidean space $\mathbb{E}^3$. Finally, we show data on Euler rotation angles
for which maximally contextual retrits (qutrits of the real Hilbert space) are
found, and derive mathematical relations through data analysis between Euler
angles and qutrit states parameterized with spherical coordinates.
