For some simple graphs when multiplied with rooted multiplication, the choice of node to mark as the root doesn't effect the result. Examples include:
The zero, one and two node graphs.
Cycles
Fully connected graphs.
These graphs seem clearly symmetrical, but is there a name for this sort of symmetry?
The meaning of rooted product: https://en.m.wikipedia.org/wiki/Rooted_product_of_graphs
Wiki also lists various symmetry like constraints such as zero-symmetric, semi-symmetric, and distance-transitive, however I'm still not sure.
It sounds like the property that you are looking for is vertex-transitive. We say that a graph $G$ is vertex transitive if for any two vertices $u$ and $v$, there exists an automorphism on $G$ (that is, an isomorphism of $G$ onto itself) that takes $u$ to $v$. In a vague sense, vertex transitivity means that no vertex of $G$ is special, that the graph looks the same from any vertex. Examples contain the graphs you stated: cycles, complete graphs, the petersen graph, etc.
If you take a rooted product of $G$ with $H$, where $H$ is vertex transitive, then it doesn't matter what you choose the root of $H$ to be, since "no vertex of $H$ is special." There may be particular cases of $G$ and $H$ where the choice of the root doesn't matter, but this would be highly dependent on the structures of $G$ and $H$. By and large, it seems, $H$ would have to be vertex transitive for the choice of the root to not affect the result of the rooted product.