Are all even graphs planar?

Recall that a graph is even when all its vertices are of even degree (number of incident edges; loops contributing two degrees each).

If not, is there any extra (set of) condition(s) needed to make an even graph planar (such as being simple, connected, etc.)

Surprisingly, I have difficulty looking up the answer, even though this seems like a simple and natural question to ask. Regrettably, I don't know enough about graph theory to figure it out by myself.

The ultimate criterion for planar graphs is Wagner's theorem, which roughly says that a graph is planar whenever it doesn't contain a copy of the complete graph on five vertices nor a copy of the complete bipartite graph on $3+3$ vertices. [The more precise statement is that a graph is planar whenever it has neither of these two graphs as a minor.] Neither of these two small graphs is planar, and the theorem basically says that if a graph doesn't have these small local reasons not to be planar than the graph itself is planar.

Notice this has very little to do with the degrees of the vertices. So for example, complete graphs on $2n+1$ vertices with $n\ge2$, or bipartite graphs on $(2m)+(2n)$ vertices with $m,n\ge2$, are even yet not planar.

K5 is not planar but every vertex has 4 edges on it.