  Hourglass dispersion is generally believed to be solely protected by
nonsymmorphic symmetries, because these symmetries can introduce
high-dimensional projective representations. Here, based on symmetry arguments,
we propose that the hourglass dispersion can be jointly protected by symmorphic
and nonsymmorphic symmetries, instead of the only conventional nonsymmorphic
symmetry. Moreover, using first-principles calculations, we realize our
proposal in phonon spectra of realistic materials that share an antiperovskite
structure with space group \emph{P}4/\emph{nmm}. Importantly, the neck points
of these hourglass dispersions trace out two nodal rings tangent to four nodal
lines, forming a unique hourglass nodal cage in the bulk Brillouin zone. The
Berry phase analysis reveal the nontrivial topology of these nodal rings and
nodal lines. Furthermore, the nontrivial surface states and isofrequency
surface arcs are visible, facilitating their experimental confirmation of such
exotic quasiparticles. Our work not only offers a new insight into the
hourglass dispersion, but also expands aspects for studying promising
topological quasiparticles in condensed-matter systems.
