  The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can
often count the fixed points of elements of a cyclic group acting on a
combinatorial set by plugging roots of unity into a polynomial related to this
set. One of the most impressive instances of the cyclic sieving phenomenon is a
theorem of Rhoades asserting that the set of plane partitions in a rectangular
box under the action of promotion exhibits cyclic sieving. In Rhoades's result
the sieving polynomial is the size generating function for these plane
partitions, which has a well-known product formula due to MacMahon. We extend
Rhoades's result by also considering symmetries of plane partitions:
specifically, complementation and transposition. The relevant polynomial here
is the size generating function for symmetric plane partitions, whose product
formula was conjectured by MacMahon and proved by Andrews and Macdonald.
Finally, we explain how these symmetry results also apply to the rowmotion
operator on plane partitions, which is closely related to promotion.
