  For their bijective proof of the hook-length formula for the number of
standard tableaux of a fixed shape Novelli, Pak and Stoyanovskii define a
modified jeu de taquin which transforms an arbitrary filling of the Ferrers
diagram with $1,2,...,n$ (tabloid) into a standard tableau. Their definition
relies on a total order of the cells in the Ferrers diagram induced by a
special standard tableau, however, this definition also makes sense for the
total order induced by any other standard tableau. Given two standard tableaux
$P,Q$ of the same shape we show that the number of tabloids which result in $P$
if we perform modified jeu de taquin with respect to the total order induced by
$Q$ is equal to the number of tabloids which result in $Q$ if we perform
modified jeu de taquin with respect to $P$. This symmetry theorem extends to
skew shapes and shifted skew shapes.
