Does an octahedron have more than $5$ reflectional symmetries? I counted $5$ planes of reflection for the octahedron: two corresponding to planes orthogonal to two sides and going through one vertex, two corresponding to planes diagonal and one corresponding to the "obvious" plane through the middle 4 vertices. I read that the octahedron should have the same number of reflections as the cube (for which I found $9$) but I see absolutely no other way of putting a plane through the octahedron so that it is a plane of reflectional symmetry. Are there really $9$ in total and if so, which ones am I missing?

For the first type you counted two planes, with a north and south pole. However there are three ways to choose north/south poles, so there should really be six reflections not two.

For the second type your plane is the equator, and again a north and south pole. But there are three ways again, so instead of one there should really be three.