Let $P_2 $ be the two disk minus 2 smaller disks in its interior.
Put $M_1 := P_2 \times \mathbb{S}^1$, $M_2 := M_1$. These are obviously Seifert fibered spaces as they have a trivial circle action. $\partial M_i$ is disjoint union of three tori, pick one and call it $T$.
Glue $M_1$ to $M_2$ along $T$ using a diffeomorphism $T\to T$ that exchanges fiber with base.
Problem: The resulting manifold $M = M_1\bigcup_T M_2$ is not a Seifert fibered manifold.
The fact is that the action of rotation of the fiber of $M_1$ restricted to the boundary becomes a rotation of the (exterior) boundary of $P_2$ in $M_2$ under the diffeomorphism. However $P_2$ is punctured twice therefore this action does not extend to an action of $M_2$.
I am trying to understand the geometry of graph manifolds like $M$. I think that in this case the torus $T$ is incompressible hence in the JSJ decomposition of $M$ we will have $M_1$ and $M_2$ as blocks.
Is this correct?