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.