I want to find some general arguments about why it can happen that $$ M \sharp \overline{M} \not\simeq M \sharp M$$ for $M$ a compact orientable odd dimensional manifold, which is chiral, i.e. ($M \not\simeq \overline{M}$). By $\overline{M}$ I mean the same manifold equipped with the reversed orientation.
In this M.O. answer They are considering $CP^2\sharp \overline{CP}^2$ and use the intersection form, but this is a specific tool for $4k$-dim manifolds, and hence I cannot use it in this case.
In this article at page $3$ they consider the so called Linking Form for odd dimensional closed manifolds, but I'm not able to find a clear definition of it anywhere, and more importantly, what are its properties.
Here Terry Tao claims that if $M$ is chiral, then $M \sharp M \not\simeq M \sharp \overline{M}$. I'm mostly interested in this case, but I cannot understand the reasoning he does.
Is there some more easy arguments to prove the claim in the case that $M$ is chiral and odd dimensional?
My first attempt was to prove that in the case that $M \sharp M \simeq M \sharp \overline{M}$ I could somewhat obtain a map from $M$ to $\overline{M}$, but I doubt that this approach will work (not easy and not true in general that I can find maps from $M$ to the connected sum). I searched for some suitable invariants but didn't found anything