Self-antiparticles and broken symmetries certain particles (i.e: certain bosons like the photon) do not have an anti-particle, or rather, they are they own anti-particles.
Let's assume that such symmetry is only approximate and these particles are actually different than their antiparticles, just that the broken symmetry is very small and hard to detect.
Question 


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*What experimental facts would we expect to change if such symmetries were only approximate?

*What physical consequences would they show?

EDIT
 The answers have focused on the argument that any symmetry breaking of this kind would be an all-or-nothing proposition. I would like to explore that argument a bit more with a more precise notion of a soft-broken symmetry, which would hopefully, elicit answers that will expose better the reason why symmetry breaking has to be an all-or-nothing:


*What about photons hypothetically having a slightly higher coupling constant to matter than to antimatter, and a symmetric situation with their dagger counterparts, which would couple slightly stronger to anti-matter?

*What argument exists to discard that possibility theoretically? 

*Maybe we have bounds on the delta of such couplings? 

*To what fraction of $\alpha$ are such bounds known experimentally?
 A: Dear lurscher, in quantum mechanics - as demonstrated in quantum field theory - particles of the same species are identical so their wave functions are symmetric (for bosons) or antisymmetric (for fermions). If your new hypothetical antiparticle species were physically different, this symmetry or antisymmetry would have to be broken, and this would not be a small change of the physics - it would be a huge and qualitative change of physics.
For example, there would have to exist two independent gravitational fields if you said that the gravitons were not the same thing as the antigravitons. In a similar way, there would be too independent electromagnetic fields - one excited by matter and one excited by antimatter (and containing photons that would be different). Those possibilities are excluded both theoretically and experimentally.
A: Within the Standard Model of particle physics it is  the gauge bosons that are their own antiparticle, Z,W, photon (gammas, xrays, light, infrared), plus the gluon of QCD . The Higgs which has recently  been discovered is also a boson and its own antiparticle . The SM is consistent with all data gathered up to now.
For any consequences from a hypothesis that there exists an antiphoton to a photon, one would need much more detail than the statement, i.e. the difference consisting in what? Quantum numbers? That there exists a higher symmetry that gives a different quantum number to gauge bosons than to gauge antibosons? One would have to posit a higher symmetry, certainly necessitated by new solid data, and then, if it predicts such a strange situation, check for existence of antiphotons according to the predictions of this new symmetry to falsify or prove its consistency .
The "anti" label has to be defined.
I cannot even think of an experiment that would distinguish between putative antiphotons from photons given the world view we have of particle physics now.
