  The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined
combinatorially as a weighted sum of lattice paths contained in certain
triangles, or algebraically as a complicated sum of rational functions indexed
by partitions of $n$. This paper proves the equivalence of the two definitions
for all $m\geq 1$ and all $n\leq 4$. We also give a bijective proof of the
joint symmetry property $C^{(m)}_n(q,t)=C^{(m)}_n(t,q)$ for all $m\geq 1$ and
all $n\leq 4$. The proof is based on a general approach for proving joint
symmetry that dissects a collection of objects into chains, and then passes
from a joint symmetry property of initial points and terminal points to joint
symmetry of the full set of objects. Further consequences include unimodality
results and specific formulas for the coefficients in $C^{(m)}_n(q,t)$ for all
$m\geq 1$ and all $n\leq 4$. We give analogous results for certain
rational-slope $q,t$-Catalan polynomials.
