The Galactic Bulge exploration V.: The secular spherical and X-shaped Milky Way bulge

In this work, we derive systemic velocities and subsequently orbits for 8456 RR~Lyrae stars. We identify interlopers from other Milky Way (MW) structures, which amount to 22 percent of the total sample. Most interlopers are associated with the halo, with the remainder linked to the Galactic disk. We confirm the previously reported lag in the rotation curve of bulge RR~Lyrae stars regardless of the removal of interlopers. Metal-rich RR~Lyrae stars' rotation patterns are consistent with that of non-variable metal-rich giants, following the MW bar, while metal-poor stars exhibit slower rotation. The analysis of orbital parameter space is used to distinguish bulge stars that, in the bar reference frame, have prograde orbits from those in retrograde orbits. We classify the prograde stars into orbital families and estimate the chaoticity (in the form of frequency drift) of their orbits. RR~Lyrae stars with banana-like orbits have a bimodal distance distribution, similar to the distance distribution seen in the metal-rich red clump stars. The fraction of stars with banana-like orbits decreases linearly with metallicity, as does the fraction of stars on prograde orbits (in the bar reference frame). The retrograde moving stars (in the bar reference frame) form a centrally concentrated nearly spherical distribution. Analyzing an $N$-body+SPH simulation, we find that some stellar particles in the central parts oscillate between retrograde and prograde orbits and only a minority stays prograde over a long period of time. Based on the simulation, the ratio between prograde and retrograde stellar particles seems to stabilize within a couple of gigayears after bar formation. The non-chaoticity of retrograde orbits and their high numbers can explain some of the spatial and kinematical features of the MW bulge that have been often associated with a classical bulge.