How many Euler tours exist in a given graph? A Euler tour is defined like that: Let $G = (V, E)$ be a graph and $C$ a circuit in $G$. $C$ is called Euler tour $\Leftrightarrow$ every edge $e \in E$ is exactly once in the circuit. If a graph $G$ has at least one Euler tour $C$ that starts with $v \in V$, can $G$ have another Euler tour that also starts with $v$ and does not simply go into the other direction?

Certainly. The usual proof that Euler circuits exist in every graph where every vertex has even degree shows that you can't make a wrong choice. So if you have two vertices of degree $4$, there will be more than one circuit. Specifically, think of $K_5$, the complete graph on $5$ vertices. Any permutation of $12345$ is a start of a Euler circuit-then hit the other edges either way around, $48$ of them starting at any given vertex. There are more, too, as $1521345241$ is another which returns to start not halfway through.