The reason this is true is because the dodecahedron is the dual of the icosahedron and vice versa (that means the corners of the dodecahedron have the same symmetry as the faces of the icosahedron). This applies to lots of other examples, for instance the Face-Turning Octahedron is the same as a corner-turning cube because they are dual to each other, or even a face-turning rhombic dodecahedron is the same as an edge-turning cube.
In fact, Ryan's Face and corner-turning icosahedron could also be viewed as either a face-turning icosidodecahedron or a corner-turning rhombic triacontahedron because the corners of the RT and the faces of the Icosidodecahedron have the same symmetry as both the faces and corners of the dodecahedron or icosahedron (that's because some of them have 3 fold symmetry, the icosahedral symmetry, and others have five-fold symmetry, the dodecahedral symmetry).
What's interesting is that there are some instances of self-duality. The tetrahedron, for example, is dual to itself, which means you could perceive a Pyraminx as both face turning and corner-turning.
I really hope this makes sense...