  We define a number of related combinatorial objects, each of which possesses
a surprising symmetry. We include several applications such as a combinatorial
explanation for certain fixed points of the involution $\omega$ on the ring of
symmetric functions, as well as a relationship between certain skew Schur
functions and skew $Q$-Schur functions. We give a $t$-deformation of these
$Q$-Schur functions, and show that it is Schur positive, including a
combinatorial description of the Schur coefficients. A corollary of our results
is the equality of skew $Q$-Schur functions: $Q_{\lambda+\delta/\mu +
\delta}=Q_{\lambda'+\delta/\mu' + \delta}$ for $\mu \subseteq \lambda$ and
$\delta=(n,\ldots,1)$ for some $n > l(\lambda)$.
