Let $p,q :(-1,1) \mapsto \mathbb{C}$. Assume the symmetry property $p(-x) = \overline{p(x)}$ and $q(-x) = \overline{q(x)}$ for $x \in (-1,1)$. Consider the Sturm-Liouville eigenvalue problem
$$(pu')' + qu = \lambda u \\ \lim_{x \to \pm1} p u' =0$$
Assume also that the eigenvalues $\lambda$ are only real. When can we say that $u$ would also satisfy the same symmetry property as the coefficients?