What can be said about pairs of non-isomorphic groups which are epimorphic images of one another and which also embed into one another?

Can such pairs of groups be 'classified' in some sufficiently weak, but still non-trivial sense, or are they just too common to hope for anything like this?

Obviously, groups forming such pairs can neither be Hopfian nor co-Hopfian.

An example of such pair of groups consists of $\rm C_\infty \times \rm F_2^\infty$ and $\rm F_2^\infty$, where $\rm C_\infty$ denotes the infinite cyclic group and $\rm F_2$ denotes the (nonabelian) free group of rank 2.