3-manifolds with isomorphic fundamental groups There are many non-homeomorphic 3-manifolds with isomorphic fundamental groups, for example the lens spaces $L(p,q_1)$ and $L(p,q_2)$ with $q_1 \ne \pm q_2^{\pm 1}\mod p$. Also, Seifert fibered spaces over the disk with two exceptional fibers can have isomorphic fundamental groups. A third type of example can be constructed by using the connect sum of knot of a chiral knot $K$, $S^3-(K\#K)$ and $S^3-(K\#-K)$ (think of the square and granny knots). Even more examples can be constructed by connect summing the examples in this paragraph (in the sense of 3-manifolds as opposed to knots). 
However, there are also non-orientable and orientable manifolds with isomorphic fundamental groups, which motivates my question:

Is there a list of all non-homeomorphic pairs of 3-manifolds (orientable and non-orientable) with isomorphic fundamental groups?

Given, the likely complexities of reducible manifolds and graph manifolds, it would be fine if it was only a list of geometric 3-manifolds. 
 A: (Edit: Everything what follows is about closed and orientable $3$-manifolds.)
Non-spherical geometric $3$-manifolds are determined by their fundamental group. This is proved in Peter Scott's paper "There are no fake Seifert fiber spaces with infinite $\pi_1$" http://www.jstor.org/stable/2006970 (He does the case of Seifert fibrations. For hyperbolic $3$-manifolds the uniqueness of course follows from Mostow rigidity.)
For irreducible manifolds with nontrivial JSJ-decomposition one can apply Waldhausen's rigidity theorem (because these manifolds are necessarily Haken) to conclude that the are also determined by their fundamental groups. 
For reducible $3$-manifolds things are a bit more complicated because one can choose orientations of the prime components separately. 
Section 2 of the survey paper of Aschenbrenner-Friedl-Wilton http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/3-manifold-groups-final-version-031115 explains why there are only three ways to get 3-manifolds with isomorphic fundamental groups.These three ways are


*

*the examples of lens spaces mentioned in the question.

*manifolds of the form $M\sharp N$ and $M\sharp \overline{N}$, where $N$ and $\overline{N}$ are the same manifold with opposite orientations

*examples constructed from the above by taking connected sums.
