Suppose you have an infinite graph $G$. I assume $G$ to be cubic and planar. No further conditions, so it will be asymmetric, maybe in the sense of cubic planar version of Rado's graph: Every possible combination of faces exists somewhere in this graph (like in normal numbers).
Now you are given a second one of such a graph $G'$. They might look different at first glance, but aren't they actually identical?