  A symmetry of $(t,q)$-Eulerian numbers of type $B$ is combinatorially proved
by defining an involution preserving many important statistics on the set of
permutation tableaux of type $B$. This involution also proves a symmetry of the
generating polynomial $\hat{D}_{n, k}(p,q,r)$ of number of crossings and
alignments, and hence $q$-Eulerian numbers of type $A$ defined by L. Williams.
By considering a restriction of our bijection, we were led to define a new
statistic on the permutations of type $D$ and $(t,q)$-Eulerian numbers of type
$D$, which is proved to have a nice symmetry as well. We conjecture that our
new statistic is in the family of Eulerian statistics for the permutations of
type $D$.
