  We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau
threefold (CY3) $X$ which is equipped with an involution $\imath$ preserving
the holomorphic volume form. We define integers $n_{g,h}(\beta) $ which give a
virtual count of the number of genus $g$ curves $C$ on $X$, in the class $\beta
\in H_{2}(X)$, which are invariant under $\imath$, and whose quotient
$C/\imath$ has genus $h$. We give two definitions of $n_{g,h}(\beta) $ which we
conjecture to be equivalent: one in terms of a version of Pandharipande-Thomas
theory and one in terms of a version of Maulik-Toda theory.
  We compute our invariants and give evidence for our conjecture in several
cases. In particular, we compute our invariants when $X=S\times \mathbb{C}$
where $S$ is an Abelian surface with $\imath (a)=-a$ or a $K3$ surface with a
symplectic involution (a Nikulin $K3$ surface). For these cases, we give
formulas for our invariants in terms of Jacobi modular forms. For the Abelian
surface case, the specialization of our invariants $n_{g,h}(\beta) $ to $h=0$
recovers the count of hyperelliptic curves on an Abelian surface first computed
by Bryan-Oberdieck-Pandharipande-Yin.
