We prove a collection of conjectures of D. White \cite{WComm}, as well as
some related conjectures of Abuzzahab-Korson-Li-Meyer \cite{AKLM} and of Reiner
and White \cite{ReinerComm}, \cite{WComm}, regarding the cyclic sieving
phenomenon of Reiner, Stanton, and White \cite{RSWCSP} as it applies to
jeu-de-taquin promotion on rectangular tableaux. To do this, we use
Kazhdan-Lusztig theory and a characterization of the dual canonical basis of
$\mathbb{C}[x_{11}, ..., x_{nn}]$ due to Skandera \cite{SkanNNDCB}. Afterwards,
we extend our results to analyzing the fixed points of a dihedral action on
rectangular tableaux generated by promotion and evacuation, suggesting a
possible sieving phenomenon for dihedral groups. Finally, we give applications
of this theory to cyclic sieving phenomena involving reduced words for the long
elements of hyperoctohedral groups and noncrossing partitions.