I'm interested in the issue of "explanatory" mathematical proofs and would like to try to find out what intuitions mathematicians have about proofs of "if and only if" claims, like this one from graph theory "a connected graph is Eulerian if and only if all of its vertices have even degree", and this example from real analysis "a sequence of real numbers is convergent if and only if it is Cauchy". I know that "explanation" and "explanatory" are very vague words, but I'm just hoping to find out what mathematicians think about a particular issue.
In a lot of "if and only if" proofs that I've come across, one direction seems to do some explanatory work, whilst the other direction seems to be "trivial" or "obvious" and so of little, if any, explanatory power. So I was wondering if anyone had an example of a particular proof of an "if and only if" claim where they felt both directions to be explanatory? If you do, I'd be very grateful if you could post them here, and possibly try to indicate what about both directions you find explanatory? Thanks in advance for your help.