  In this paper, we provide both a preservation and breaking of symmetry
theorem for $2\pi$-periodic problems of the form \begin{align*} \begin{cases}
-u''(t) + g(u(t)) = f(t)\cr u(0) - u(2\pi) = u'(0) - u'(2\pi) = 0 \end{cases}
\end{align*} where $g: \mathbb{R} \to \mathbb{R}$ is a given $C^1$ function and
$f: [0,2\pi] \to \mathbb{R}$ is continuous. We provide a preservation of
symmetry result that is analogous to one given by Willem (Willem, 1989) and a
generalization of the theorem given by Costa-Fang (Costa and Fang, 2019). Both
of these theorems use group actions that are not normally considered in the
literature.
