  We derive from an eigendielectric displacement vector summation approach the
frequency order of the polar phonon modes and the polarization-dependent
structure of the reststrahlen bands within the monoclinic plane of materials
with monoclinic crystal symmetry. We show that phonon modes occur in associated
pairs of transverse and longitudinal optical modes, and that pairs either
belong to inner or outer phonon modes. Inner mode pairs are nested within outer
mode pairs. Outer mode pairs cause polarization-dependent reststrahlen bands.
Inner mode pairs cause polarization-independent reststrahlen bands. The
directional limiting frequencies within the Born-Huang approach are bound to
within outer mode regions not occupied by inner mode pairs. Hence, an unusual
phonon mode order can occur where both lower-frequency as well as
upper-frequency limits for the directional modes can be both transverse and/or
longitudinal modes. We exemplify our findings using experimental data for the
recently unraveled case of monoclinic symmetry $\beta$-Ga$_2$O$_3$ [Phys. Rev.
B~\textbf{93}, 125209 (2016)] and demonstrate excellent agreement with results
from density functional theory calculations.
