  We use the conjugate angle of radial action ($\theta_R$), the best
representation of the orbital phase, to explore the "mid-plane, north branch,
south branch" and "Monoceros area" disk structures that were previously
revealed in the LAMOST K giants (Xu et al. 2020). The former three
substructures, identified by their 3D kinematical distributions, have been
shown to be projections of the phase space spiral (resulting from
nonequilibrium phase mixing). In this work, we find that all of these
substructures associated with the phase spiral show high aggregation in
conjugate angle phase space, indicating that the clumping in conjugate angle
space is a feature of ongoing, incomplete phase mixing. We do not find the
$Z-V_Z$ phase spiral located in the "Monoceros area", but we do find a very
highly concentrated substructure in the quadrant of conjugate angle space with
the orbital phase from the apocenter to the guiding radius. The existence of
the clump in conjugate angle space provides a complementary way to connect the
"Monoceros area" with the direct response to a perturbation from a significant
gravitationally interactive event. Using test particle simulations, we show
that these features are analogous to disturbances caused by the impact of the
last passage of the Sagittarius dwarf spheroidal galaxy.

Astrophysics of Galaxies