Harvey Friedman at the 1974 ICM motivated Reverse Mathematics by the following statement:
When the theorem is proved from the right axioms, the axioms can be proved from the theorem.
Reverse Mathematics has had many successes in finding the "right axioms," but to date mainly for theorems of classical analysis, where real numbers (or equivalent infinite objects, such as sets of natural numbers) are involved. This may be partly for historical reasons, since the subject grew from the study of subsystems of second order number theory.
It seems to me that there are classical theorems in other fields that also fit the Reverse Mathematics paradigm: add a strong axiom $A$ to a weak theory $W$ and prove $A$ equivalent, over $W$, to a strong theorem $T$ of $W+A$. The example I have in mind is where $W$ is the theory of a projective plane $P$, which has three simple axioms about objects called "points" and "lines":
Any two points belong to a unique line.
Any two lines have a unique common point.
There exist at least four distinct points.
This $W$ is a weak theory with very few interesting theorems. But if we add to $W$ the theorem of Pappus as an axiom $A$, then results of von Staudt, Hilbert, and Hessenberg enable us to prove the theorem
$T$: The plane $P$ can be coordinatized by a field.
Conversely, if $T$ holds then we can prove $A$. This is because the Pappus axiom $A$ states that certain points lie on a line, and $A$ can be proved once we have coordinates in a field -- by computing the coordinates of the points in question and showing that they satisfy a linear equation.
There are a few variations of this result. For example, if axiom $A$ is replaced by the theorem of Desargues then the equivalent theorem $T$ of $W+A$ is that $P$ can be coordinatized by a skew field.
This leads me to the following questions. Are these results reasonable examples of Reverse Mathematics? Are there other natural examples outside analysis?
Edit (Nov 15, 2014). Many thanks to the logicians who have answered this question. It appears to me now that the term "reverse mathematics" is too narrow for what I had in mind, but I am still interested in examples of "finding the right axioms." An even more elementary example than the Pappus axiom/theorem is the Euclid's parallel axiom. It is needed to prove many theorems, e.g. Pythagorean theorem, angle sum of a triangle equals $\pi$, ... and these theorems also prove the axiom.