Does energy have a sign? Quantities like position and time allow us to place our origin anywhere, but can the same be said for energy? I was thinking about the way we have defined the quantity gravitational potential, and for any finite distance from a 'planet' or a body, an abject would be said to have negative gravitational potential, conveying that a negative amount of work is done (in bringing the object from 'infinity' to that position, but this seems impossible as negative energy does not seem to physically exist (as far as I know). So, does negative energy really exist, or does it not (implying that negative work done is just another mathematical argument)?
Edit: The primary reason I ask this question is because I was wondering whether negative work done is fundamentally different from negative debts (when talking about money), or are they just the same - a mathematical convenience.
 A: The question about "existence" is philosophical one, not a physics one. Some people would argue that you can't be sure about the existence of anything, including yourself. 
Physics is concerned with creating mathematical models that can make predictions about the outcome of a experiment or situation. All physical models have limited applicability, i.e. the predictions are only accurate if certain conditions are met. So far there aren't any models that are truly universal and applicable in every possible situation. 
Negative, or positive, energy are just that: concepts that are useful to make quantitative predictions in certain situations (but not in others). 
For example, negative energy is a useful concept is energy is moving around inside a system. If it's moving one way you call it "positive" , if it's moving the other way it's negative. 
A: Of course it is very important in some calculations if the energy change in some system is positive or negative. If you are building a power plant, rather try to get a positive amount of energy out!
But that's only differences in energy. The point of energy zero is defined arbitrarily and you can always redefine it, so that the absolute sign of $E$ is unimportant.
A: There's no negative energy, and the origin for energy is absolute, not arbitrary. 
To measure the energy of a system, you can simply measure its mass, then use $E = mc^2$ to find its energy (in a frame where its net momentum is zero). The energy is always positive because the mass is always positive. Maybe you can find some really exotic counterexamples with the Casimir effect or something, but basically it's positive.
When we say that the gravitational energy for the Earth-moon system is negative, it just means that the energy is lower than it would be if the moon were further out. The Earth-moon system has a certain mass. It's less than the mass of the Earth plus the mass of the moon, each considered separately. This is called the "mass defect". 
So we say there's negative energy in the gravitational binding of them to each other, but that's just so that the total sum of energy comes out right. The total amount of energy is still positive because the mass of the system is positive. 
When people say that the origin of potential energy doesn't matter, they mean that forces are derivatives of the potential energy. If you add a constant to the potential energy, that doesn't change any derivatives, so if all you care about is finding the forces, it doesn't matter if you add a constant to the potential energy. However, there is still only one choice of zero for the energy of an entire system that accounts for all the energy, and that's one that makes $E = mc^2$ true.
Note: this presumes we're dealing with systems that have a rest frame; massless systems don't have a rest frame.
A: To be honest, signs don't exist. Energies don't exist. They are merely methods or models or descriptions invented for us to describe forces, tendencies, absorption of radiation, impacts, vibrational motions, directions etc. So, does negative energy exist? We can just invent it, like you just described for the potential energy, so yes.


*

*In some situations a sign describes nothing but the size of values (when it doesn't matter where the origin is, such as with potential energies with arbitrarily chosen reference points),

*in other situations the sign describes the mathematical act of adding or subtracting (like heat absorbed or expelled),

*while in yet other situations signs describe directions on a predefined axis (your axis defines forward, so a speed moving you backwards is negative).

*Etc.


A sign is a mathematical invention that means different things in different contexts, depending on what we need to do.

conveying that a negative amount of work is done

A "negative amount of work" belongs to the 2nd bullet point above. Work being negative just means that energy is leaving (mathematically subtracted) the object. The negative sign is here nothing more than mathematical.
A: Energy is a concept which can be positive or negative.  However, it is interesting in that, in many situations, it is meaningful to define a "minimum" energy.  This is different than other things, like positions.  There is rarely a meaningful "minimum position," unless there is a symmetry which makes such a concept meaningful.
Because there is a minimum, it can be meaningful to assign it to the value 0.  We could have picked any value to assign it to, but zero is convenient.  If we make such an assignment, then it is trivial to show that energy must always be non-zero.
A: The energy density equations for an electric field for a capacitor with a dielectric contain a negative component. This energy has to be supplied to pull the dielectric of a capacitor. Unlike most negative energy components this one has neither a floating base (like most potential energies) nor an obvious rule that keeps the total energy positive. So far, no attempts have succeeded to make a dialectic low enough mass and charge the capacitor to a high enough voltage so that the negative energy component exceeds the mass energy of the dielectric. If this ever succeeded (and I know of no law that absolutely forbids it though it seems unlikely to work) then negative energy components would be realized and we could hold wormholes open.
The consequence for constructed negative energy in physics is backwards time travel.
A: The answers so far to this question have been both irrelevant and wrong. The question isn't some philosophical nonsense about whether negative energy really exists or whether energy really exists. It is a straightforward question, about whether there is a shift-invariance in energy, i.e. if the energy of the universe increased by a fixed quantity, would there be an impact on the physics?
The answer is clearly no, there is no impact, you do have "energy translational" invariance. The reason this is not talked about is that it's completely useless. It makes no sense to increase the energy of every object by the same amount (because then the system "A + B" will have to increase its energy by 2E, not by E), so what you're really talking about is a shift in your unit system, where you call "0 J"  as "20 J" instead, and then energy is not easily additive anymore (you have to subtract 20 every time you add things). This is like using Celsius instead of Kelvin, or pH instead of proton concentration, there is no physical importance to this.
Note that it is impossible to have any unit system in which negative energy doesn't exist, as you can get arbitrarily negative gravitational potential energies near a black hole, and to translate -∞ to 0, you need to map everything else to +∞.
Edit -- Global shifts in local energy density are meaningful to speak of, since it's a local phenomenon and A + B doesn't need to shift by the same amount as A and B. In this case, the physics obviously isn't invariant, since you get more gravity.

Answering the question that wasn't asked, but everyone else is answering --
Yes, energy does exist, it means nothing to say that something is "just a mathematical construct". You can define energy positivistically, and measure it, this is fundamentally no different from measuring a chair, which also means you're measuring the light from the chair incident on your eyes, etc.
