  The \emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$
that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials
have important connections to representation theory, algebraic geometry, and
symmetric functions. Haglund and Haiman discovered combinatorial formulas for
$C_n(q,t)$ as weighted sums of Dyck paths (or equivalently, integer partitions
contained in a staircase shape). This paper undertakes a combinatorial
investigation of the joint symmetry property $C_n(q,t)=C_n(t,q)$. We conjecture
some structural decompositions of Dyck objects into "mutually opposite"
subcollections that lead to a bijective explanation of joint symmetry in
certain cases. A key new idea is the construction of infinite chains of
partitions that are independent of $n$ but induce the joint symmetry for all
$n$ simultaneously. Using these methods, we prove combinatorially that for
$0\leq k\leq 9$ and all $n$, the terms in $C_n(q,t)$ of total degree
$\binom{n}{2}-k$ have the required symmetry property.
