Can anyone explain to a novice physicist whether there is a gravitational-electromagnetic symmetry? I am trying to understand how the four fundamental forces relate to one another and to a theory of everything. As I understand it the unified force that is thought to exist at very high energies gets broken via symmetry breaking into the four forces observed at our energy. I don't fully understand how this happens and in particular if there is a gravitational- electromagnetic symmetry and if, when and how this breaks. 
 A: We only have experimental evidence for one symmetry breaking i.e. the breaking of the electroweak force into seperate electromagnetic and weak forces. With the discovery of the Higgs boson the electroweak symmetry breaking is now well established.
The mechanism by which symmetry breaking occurs is complex, and to be honest it's impossible to give a layman an accurate idea of what is going on, though you'll find no end of attempts in the popular science media. If your aim is to fully understand the process then I'm afraid you're in for a lot of work. The best semi-rigorous description I've seen is on Matt Strassler's blog, though even this will be intimidating for beginners.
Anyhow, there is some evidence that the electromagnetic, weak and strong forces converge to similar strengths at an energy scale of around $10^{16}$ GeV, and this has lead to speculation about a symmetry breaking similar to the electroweak symmetry breaking, but happening at much higher energies and splitting the strong and electroweak forces. The best known models for this are probably the Georgi–Glashow SU(5) theory and the SO(10) theory. However there is no experimental suppoprt for these theories so at the moment they are entirely speculative. Actually I think proton lifetime measurements have ruled out the simplest SU(5) theory, though I confess I'm not up to speed with this area.
Anyhow, give that physicists have conjectured unification of the electromagnetic, weak and strng forces it seems an obvious extension to consider unification of the gravitational force as well. However we have no theory of quantum gravity, and unification of gravity with the other forces is highly speculative. String theory is probably the best contender for such a unified theory, but it remains speculative.
Now, you started by asking about unification of gravity and the electromagnetic force. Assuming the above descripton bears any connection to reality, gravity and EM have not been unified since about the Planck energy. At that time gravity split from the strong-electroweak force. Since then two more splittings have produced the separate EM force we see today. So there is no gravitational-electromagnetic symmetry in any useful sense.
As a historical note: in 1919 Theodore Kaluza discovered a possible unification of gravity and the EM force based on extending general relativity to a five dimensional spacetime. However this quickly proved unrealistic and today is regarded as an interesting but irrelevant piece of the history of physics. Though having said that, the basic idea of extending physics to extra dimensions is alive and well.
