  In this paper we study para-Hermitian rational matrices and the associated
structured rational eigenvalue problem (REP). Para-Hermitian rational matrices
are square rational matrices that are Hermitian for all $z$ on the unit circle
that are not poles. REPs are often solved via linearization, that is, using
matrix pencils associated to the corresponding rational matrix that preserve
the spectral structure. Yet, non-constant polynomial matrices cannot be
para-Hermitian. Therefore, given a para-Hermitian rational matrix $R(z)$, we
instead construct a $*$-palindromic linearization for $(1+z)R(z)$, whose
eigenvalues that are not on the unit circle preserve the symmetries of the
zeros and poles of $R(z)$. This task is achieved via M\"{o}bius
transformations. We also give a constructive method that is based on an
additive decomposition into the stable and anti-stable parts of $R(z)$.
Analogous results are presented for para-skew-Hermitian rational matrices,
i.e., rational matrices that are skew-Hermitian upon evaluation on those points
of the unit circle that are not poles.
