I am thinking of an example where the connected sum of two three Manifolds depends on the chosen orientation. Hempel gives in his book "3-Manifolds" an example, namely lens spaces. He shows that two lens spaces
$L_{p,q}, L_{p,q'}$ are homeomorphic $\iff qq'$ is congruent to $\pm 1$ modulo $p$. However, he only shows necessity in his book, and says, the desired homeomorphism can be obtained by construction. Now, I cant imagine how to write down explicitly such a homeomorphism.
I hope you can help me.