Assume all manifolds involved to be closed and orientable.
During my studies I learnt about the signature of a $4k$-manifold, and it turned out that (when it is non zero) it can be use to make a distinction between $M$ and $-M$, where by $-M$ I mean the manifold $M$ with the opposite orientation. If $M$ is even-dimensional we can make use of the Euler class of the tangent (assuming we are working in the smooth category).
So I was wondering if there are other invariants which can detects the flip of orientation even in an odd dimensional manifold, but I found nothing on my books-notes. After some googling the word "linking form" came out, but again, I wasn't able to retrieve any clear reference to it. This question came out because I was wondering myself what can happen, (in a topological-geometrical viewpoint) when I flip the orientation of a manifold.