The authors consider the dynamics of entire functions which are symmetric with respect to some ray. More precisely, they consider maps \(f\) for which there exist a complex number \(a\) and a real \(\theta\) such that \(e^{(n-1)\theta i}(f(z)-a)^{(n)}|_{z=a}\geq 0\) for all \(n\geq 0\). They identify a point \(z_0=t_0e^{i\theta}+a\) such that if \(x>t_0\) and \(n\geq 1\), then any compact connected set containing \(z:=xe^{i\theta}+a\) and \(f^n(z)\) intersects the Julia set. They also prove a number of other results on the dynamics of \(f\). They apply these results to functions of the form \(\lambda\sinh z\) or \(\lambda \sin z\), and various other functions.