Each node has 4 edges, so is visited twice. We start and end on node A, but it must be visited some time in between as well.
Let's split this up into cases depending on how many other nodes are visited before we come back to A the first time.
A node cannot be travelled back to immediately because that reuses the same edge, so between two visits to a node at least two other nodes are visited.
1. Node A is visited after 4 other nodes, i.e. $A....A....A$. There can be no repeats in the two halves (they must be at least 2 apart, so it would have to be $AX..XA$, but that repeats edge $AX$). So all four other nodes are visited in some order in the first half. There are $4!=24$ such orderings. The second half of the route is then fixed, apart from the direction. This gives $48$ such routes.
2. Node A is visited after 5 other nodes, i.e. $A.....A...A$. There must be exactly one repeat in that first section, so $AX..X.A$ or its reverse. Any more repeats or putting the repeat further apart will cause edges to be used twice. The blanks are then the other 3 nodes in any order. This means there are $4!*2=48$ choices for this section. The second half is again fixed apart from direction, so we have $96$ such routes.
3. Node A is visited after 6 other nodes, i.e. $A......A..A$. There must be exactly two repeats in that first section. The only patterns that work without repeating any edges are $AXY.X.YA$, its reverse, and $AX.YX.YA$. The blanks are then the other 2 nodes. This means there are $4!*3=72$ choices for this section. The second half is again fixed apart from direction, so we have $144$ such routes.
4. Node A is visited after 3 other nodes. This is case 2 in reverse, so also $96$ routes.
5. Node A is visited after 2 other nodes. This is case 3 in reverse, so also $144$ routes.
This gives a total of:
$48+2*96+2*144 = 528$ routes.
More explicitly, the routes are:
Using the digits $1$-$4$ to stand for the letters $B$-$E$ in any order:
1. $A1234A2413A$, $A1234A3142A$
2. $A12314A243A$, $A12314A342A$, $A21341A423A$, $A21341A324A$
3. $A123142A34A$, $A123142A43A$, $A132412A34A$, $A132412A43A$, $A132142A34A$, $A132142A43A$
4. $A243A12314A$, $A342A12314A$, $A423A21341A$, $A324A21341A$
5. $A34A123142A$, $A43A123142A$, $A34A132412A$, $A43A132412A$, $A34A132142A$, $A43A132142A$