Bodies of constant width?
I found the reference I was looking for. The full list of cases under which $K$ is a rotor in a cavity shaped like the polytope $P$ is available on page 27 of the notes title "The use of spherical harmonics in convex geometry" by Rolf Schneider. They are available under "Course Materials" on his website. As I recall, there is one more non-trivial case in $d=3$ if the cavity is allowed to be open (e.g. a cone), and this case appears in the more complete list in Groemer's book.
This adds nothing to Yoav Kallus' answer, but I was curious to see what these rotors look like
(Schneider's notes has no figures).
I found grainy photos of rotors for the cube, the regular octahedron, and the regular tetrahedron
in a 50-year old paper by
Michael Goldberg,
"Rotors in Polygons and Polyhedra,"
Mathematics of Computation, Vol. 14, No. 71 (July, 1960), pp. 229-239:
(They remind me of stones found on a beach!)
Of course one can find much better examples of cube rotors, which as Anton points out, are just constant-width bodies. E.g., this is from the cover of Bryant and Sangwin's 2008 How Round Is Your Circle:
