  The $q,t$-Catalan number $\mathrm{Cat}_n(q,t)$ enumerates integer partitions
contained in an $n\times n$ triangle by their dinv and external area
statistics. The paper [LLL18 (Lee, Li, Loehr, SIAM J. Discrete Math. 32(2018))]
proposed a new approach to understanding the symmetry property
$\mathrm{Cat}_n(q,t)=\mathrm{Cat}_n(t,q)$ based on decomposing the set of all
integer partitions into infinite chains. Each such global chain
$\mathcal{C}_{\mu}$ has an opposite chain $\mathcal{C}_{\mu^*}$; these combine
to give a new small slice of $\mathrm{Cat}_n(q,t)$ that is symmetric in $q$ and
$t$. Here we advance the agenda of [LLL18] by developing a new general method
for building the global chains $\mathcal{C}_{\mu}$ from smaller elements called
local chains. We define a local opposite property for local chains that implies
the needed opposite property of the global chains. This local property is much
easier to verify in specific cases compared to the corresponding global
property. We apply this machinery to construct all global chains for partitions
with deficit at most $11$. This proves that for all $n$, the terms in
$\mathrm{Cat}_n(q,t)$ of degree at least $\binom{n}{2}-11$ are symmetric in $q$
and $t$.
