A family of such functions is given by 
$$\phi(x)=\phi_c(x):=e^{t_cx+cx^2}$$
for all real $x$, where $c\in(-\frac{1+\sqrt2}2,\frac{-1+\sqrt2}2)$ and $t_c=\pm\sqrt{1 - 8 c + 12 c^2 + 16 c^3}$; then both expected values equal ${e^{{2 t^2}/(1-4 c)}}/{\sqrt{1-4 c}}$.