  The Painlev\'e equations possess transcendental solutions $y(t)$ with special
initial values that are symmetric under rotation or reflection in the complex
$t$-plane. They correspond to monodromy problems that are explicitly solvable
in terms of classical special functions. In this paper, we show the existence
of such solutions for a $q$-difference Painlev\'e equation. We focus on
symmetric solutions of a $q$-difference equation known as
$q\textrm{P}_{\textrm{IV}}$ or $q{\rm P}(A_5^{(1)})$ and provide their symmetry
properties and solve the corresponding monodromy problem.
