  We study a refinement of the $q,t$-Catalan numbers introduced by Xin and
Zhang (2022, 2023) using tools from polyhedral geometry. These refined
$q,t$-Catalan numbers depend on a vector of parameters $\vec{k}$ and the
classical $q,t$-Catalan numbers are recovered when $\vec{k} = (1,\ldots,1)$. We
interpret Xin and Zhang's generating functions by developing polyhedral cones
arising from constraints on $\vec{k}$-Dyck paths and their associated area and
bounce statistics. Through this polyhedral approach, we recover Xin and Zhang's
theorem on $q,t$-symmetry of the refined $q,t$-Catalan numbers in the cases
where $\vec{k} = (k_1,k_2,k_3)$ and $(k,k,k,k)$, give some extensions,
including the case $\vec{k} = (k,k+m,k+m,k+m)$, and discuss relationships to
other generalizations of the $q,t$-Catalan numbers.
