in 3D "gyrated" is easy to define: take off part of the shape (usually a cupola), rotate it and put it back.
however in 4d+ it's gets harder, take a look at these gyrations:
1)take a runcinated tesseract and remove a cube cupola off the shape and replace it with a octahedron cupola
2) take a cube cupola and take a "square orthobicupolic ring" of it and replace it with a "square gyrobicupolic ring"
3) take an octahedral prism and cut it in two (I haven't figured out what the two halves are yet) and rotate one and put it back, (this is listed on Klizting's paper as "reflected orthogonal trigon||trigon"
4) take a runcinated 5-cell and remove a tetrahedron cupola off the shape and rotate it and put it back
in 3d the term can be defined as either "replacing a facet (face in this case) with its dual" or "rotating a face"
1) demonstrates the former while 3) represents the latter and 4) represents both. 2) represents "dualing/rotation*" face (which is a lower element
so maybe the definition should be
Take an element, not necessary a facet, and rotate** or dual it
*for a polygon these two mean the same thing.
**assuming that in "2)" there will be a triangular prism that gets rotated! (I'll need to check that:D)