Given a determinantal representation by means of a cyclic weighted shift matrix, one can show the resulting polynomial is hyperbolic and invariant under the action of the dihedral group. Chien and Nakazato (2015) asked the converse question. By properly modifying a determinantal representation construction of Dixon (1902), we show for every hyperbolic polynomial with dihedral invariance there exists a determinantal representation admitted via some cyclic weighted shift matrix with real entries.