  Let $\delta=(\delta_1,\ldots,\delta_n)$ be a string of letters $h$ and $v$.
We define a Young tableau to be $\delta$-semistandard if the entries are weakly
increasing along rows and columns, and the entries $i$ form a horizontal strip
if $\delta_i=h$ and a vertical strip if $\delta_i=v$. We define
$\delta$-promotion on such tableaux via a modified jeu-de-taquin. The first
main result is that $\delta$-promotion has period $n$ on rectangular
$\delta$-semistandard tableaux, generalizing the results of Haiman and Rhoades
for standard and semistandard tableaux. The second main result states that the
set of rectangular $\delta$-semistandard tableaux for fixed $\delta$ and
content $\gamma$ exhibits the cyclic sieving phenomenon with the generalized
Kostka polynomial. To do so we follow Fontaine-Kamnitzer and associate to
$(\delta,\gamma)$ an $SL_m$-invariant space
Inv$(V_{\lambda^1}\otimes\cdots\otimes V_{\lambda^n})$ where each
$V_{\lambda^i}$ is an alternating or symmetric representation. We show that the
Satake basis of the corresponding invariant space is indexed by the set of
tableaux corresponding to $(\delta,\gamma)$ and is permuted by rotation of
tensor factors. We then diagonalize the rotation action using the fusion
product. This cyclic sieving generalizes the result of Rhoades, and of
Fontaine-Kamnitzer (in type A), and is closely related to that of Westbury.
