What if we now forget about the original axioms, take a set of these theorems and pronounce them new axioms, that is, just assume them to be true? Would we be able to prove the original axioms using the new ones?
Not in general. Simple example. My 'axiom' is $P \land Q$, and I derive 'theorem' $P$. Well, from $P$ alone I cannot go back to $P \land Q$.
OK, you say, but I could also derive $Q$ from $P \land Q$, and from $P$ and $Q$ I can go back to $P \land Q$. Sure, but now the question becomes: what is a theorem? Because if a theorem is simply whatever logically follows from the axioms, then all axioms themselves (or at the very least something logically equivalent) will be theorems, and so of course we can derive the old axioms from the old theorems/new axioms. So if we look at theorems that way, we would obtain a trivial result.
OK, so it would be more interesting if we derive some small set of theorems from a small set of axioms, and are then able to reverse that. Do we have examples of that? Sure, we have many different axiomitizations of propositional logic, for example, each consisting of a small set of axioms (e.g the Hilbert system, Rosser system, Frege's system, etc.), and we can derive each axiom set from each other.
Indeed, if you can reverse the process in the way you indicate, then wat that means is that apparently you can have multiple different 'bases' for your theory, and that could certainly be very interesting, as it would 'swap' our conceptual understanding of what is more 'fundamental'. As another example, we can use Newtonian laws to derive that heavier objects don't fall faster to Earth than lighter ones (disregarding friction) ... but what if we take that result as our basis of physical mechanics? Then suddenly we may obtain quite a different outlook on the physical world, even if the results are the same in the end.