  The observation of giant Rydberg excitons in cuprous oxide
$\left(\mathrm{Cu_{2}O}\right)$ up to a principal quantum number of $n=25$ by
T.~Kazimierczuk \emph{et al.} [Nature \textbf{514}, 343, (2014)] inevitably
raises the question whether these quasi-particles must be described within a
multi-polariton framework since excitons and photons are always coupled in the
solid. In this paper we present the theory of exciton-polaritons in
$\mathrm{Cu_{2}O}$. To this end we extend the Hamiltonian which includes the
complete valence band structure, the exchange interaction, and the central-cell
corrections effects, and which has been recently deduced by F.~Schweiner
\emph{et al.} [Phys.~Rev.~B \textbf{95}, 195201, (2017)], for finite values of
the exciton momentum $\hbar K$. We derive formulas to calculate not only dipole
but also quadrupole oscillator strengths when using the complete basis of
F.~Schweiner \emph{et al.}. Very complex polariton spectra for the three
orientations of $\boldsymbol{K}$ along the axes $[001]$, $[110]$, and $[111]$
of high symmetry are obtained and a strong mixing of exciton states is
reported. The main focus is on the $1S$ ortho exciton-polariton, for which
pronounced polariton effects have been measured in experiments. We set up a
$5\times 5$ matrix model, which accounts for both the polariton effect and the
$K$-dependent splitting, and which allows treating the anisotropic polariton
dispersion for any direction of $\boldsymbol{K}$. We especially discuss the
dispersions for $\boldsymbol{K}$ being oriented in the planes perpendicular to
$[1\bar{1}0]$ and $[111]$, for which experimental transmission spectra have
been measured. Furthermore, we compare our results with experimental values of
the $K$-dependent splitting, the group velocity, and the oscillator strengths
of this exciton-polariton.
