We focus on copositive and completely positive cones over symmetric cones of rank at least $5$, and in particular investigate whether these cones are spectrahedral shadows. We extend known results for nonnegative orthants of dimension at least $5$ to general symmetric cones of rank at least $5$. Specifically, we prove that when the rank of a symmetric cone is at least $5$, neither the copositive nor the completely positive cone over it is a spectrahedral shadow. We then generalize the Horn matrix to the setting of symmetric cones of rank at least $5$ by introducing Horn transformations and analyzing their geometric and algebraic properties. We show that Horn transformations generate exposed rays of copositive cones over symmetric cones. We also show that Horn transformations evade the zeroth level of a sum-of-squares inner-approximation hierarchy for copositive cones over symmetric cones.