A group $G$ is is abelian if
$$ ab = ba $$
for all pairs of elements $a,b \in G$, i.e., if multiplication commutes. Another way of stating this condition is that the identity map $\mathrm{id}$ gives an isomorphism between left- and right- multiplication. In other words, we can think of left-and right- multiplication as group actions
$$L: G \rightarrow \mathrm{Sym}(G);\ L(a)\cdot b := ab, $$
$$R: G \rightarrow \mathrm{Sym}(G);\ R(a)\cdot b := ba. $$
In an abelian group we have $L(a)\cdot b = ab = ba = R(a)\cdot b$. Hence,
$$L(\mathrm{id}(a))=R(a).$$
One might consider a weaker condition: there is some isomorphism $\varphi:G \rightarrow G$ (not necessarily the identity) such that
$$L(\varphi(a))=R(a).$$
for all $a \in G$. (In other words, if we really didn't like multiplying on the right, we could always multiply on the left by $\varphi(a)$ instead without losing anything.)
Is there a standard name for this weaker condition? When can it be satisfied (if ever)? For some standard examples (e.g., the quaternions) I can work out that no such isomorphism exists, but I don't know what to say about the general case.