Count polyomino bisections
An polyomino is a non-empty connected subset of the square tiling consisting of squares joined along their edges. We will not require that polyominos be simply connected, that is they can have holes.
A bisection of a polyomino $X$ is a pair of polyominos such that they can be joined together without overlap to form $X$.
XXXX XXX X
Can be bisected into:
X XXX X
since they can be joined as follows:
1112 222 2
Your task is to take a polyomino as input and output the number of distinct bisections which can be made. Two bisections are distinct if the border between the parts is distinct. So
Are distinct bisections even though they share a rotational symmetry. But
are not distinct because the only difference is how you label the two parts.
This is code golf the goal is to minimize the size of your source code as measured in bytes
X -> 0 XX -> 1 XXX -> 2 XXXX -> 3 XXXX -> 4 X XXXX -> 8 XX XX -> 6 XX XX -> 6 X X XXX XXX -> 28 X X XXX