  Let $A$ be a general expansive matrix and let $X$ be a ball quasi-Banach
function space on $\mathbb R^n$, which supports both a Fefferman--Stein
vector-valued maximal inequality and the boundedness of the powered
Hardy--Littlewood maximal operator on its associate space. The authors first
introduce some anisotropic ball Campanato-type function spaces associated with
both $A$ and $X$, prove that these spaces are dual spaces of anisotropic Hardy
spaces $H_X^A(\mathbb R^n)$ associated with both $A$ and $X$, and obtain
various anisotropic Littlewood--Paley function characterizations of
$H_X^A(\mathbb R^n)$. Also, as applications, the authors establish several
equivalent characterizations of anisotropic ball Campanato-type function
spaces, which, combined with the atomic decomposition of tent spaces associated
with both $A$ and $X$, further induces their Carleson measure
characterizations. All these results have a wide range of generality and,
particularly, even when they are applied to Morrey spaces and Orlicz-slice
spaces, some of the obtained results are also new. The novelties of this
article are reflected in that, to overcome the essential difficulties caused by
the absence of both an explicit expression and the absolute continuity of
quasi-norm $\|\cdot\|_X$, the authors embed $X$ under consideration into the
anisotropic weighted Lebesgue space with certain special weight and then fully
use the known results of this weighted Lebesgue space.
