Is there a name for "co-normal" subgroups? I have in mind a situation where a group $G$ has two subgroups $H$ and $K$ with the property that each of them is closed under conjugation by elements of the other.
More formally: for $h \in H$ and $k \in K$ then $h^{-1} k h \in K$ and $k^{-1}h k \in H$.
Is there a name for this?

Never seen this situation explored and don't think there's a name for it. Let's say that $(H,K)$ is a conormal pair if they satisfy the property you consider.
Two elementary observations are:
Every subgroup belongs to a conormal pair since $(K,\{e_G\})$ is conormal for every $K<G$. So, given $K$ one may ask about the (?) largest $H$ such that $(H,K)$ is conormal.
If $G=HK$ and $(H,K)$ is conormal, then both $H$ and $K$ are normal in $G$.
