I've played with this idea for years, and haven't really been able to eliminate it. So, perhaps someone here can point to simple experimental evidence that would do so.
Here's the issue: Antimatter of course annihilates ordinary matter, but the more precise statement is that antiparticles annihilate the same types of particles.
That means that you can mix antimatter with matter without it going boom, as long as you only allow one of each type in the mix. For example, positrons (anti-electrons) can pass through interior of a neutron at low speed without causing annihilations for the simple reason quarks and electrons cannot pair off for annihilation.
All of which leads to this delightfully non-standard line of thought: If what we call electrons were actually anti-particles, would we even be able to tell?
A more precise way to express that is this: If all negatively charged particles had the internal chirality of antimatter, and all positively charged particles that of matter, would the resulting transformed theory lead to paradoxes or contradictions with the Standard Model? Or would it instead produce an experimentally isomorphic theory that describes what we see just as accurately as the Standard Model?
(The neutrinos of course have no charges, so their affiliations would be determined by their associations with the electron types.)
So my question is this: Would reclassifying the chirality of particles so that "pro" always means positive particles and "anti" always means negative particles result in a version of the Standard Model that is experimentally isomorphic to known results, or would it instead produce one or more clear contradictions or paradoxes?
One final note that's too much fun not to mention: a workable pro="+"/anti="-" theory would convert the mystery of why matter dominates over antimatter into a mystery of the particle mix came out so asymmetrically.
Best Answer
There is some freedom in deciding which particle in a particle-antiparticle pair is called "matter" and which is called "antimatter" but the freedom is smaller than you think. A basic problem is that your sentence
isn't really true. In fact, the opposite statement, while inaccurate, is much closer to the truth: matter and antimatter often can annihilate even if they belong to different species.
For example, a proton will rapidly annihilate with an antineutron (or an up-quark with anti-down-quark, if we look at the same process at the quark level), leaving some positron and neutrino (whose rest mass is much lower than the rest mass of either proton or antineutron) with lots of energy.
The up-quark and down-quark are different species or flavors but it would make no sense to call one of them "matter" and the other "antimatter" because they can "almost annihilate" to "almost nothing". More generally, all quark flavors are similar and it's better to call all of them "matter", especially because they may be related by symmetries that don't have a reason to include charge conjugation C.
Now, the atoms are composed of protons and electrons – and we call the atomic bound states "matter". That implies that an electron has the same "pro-anti" label as the six quarks. There are no bound states between positrons and protons so there would be no atomic "matter" if you flipped the convention for electrons but not protons.
Grand unified theories actually do link some 2-component spinors to larger representations and these representation contain fields that create both matter and antimatter so the binary label "pro-anti" becomes more subtle in such theories. We must still carefully distinguish a field and its Hermitian conjugate.
The "pro-anti" dichotomy remains meaningless for some particles, anyway. There are totally neutral particles – photons, Z-bosons, gluons, gravitons – that are identical to their antiparticles so here there is no "polarization", of course. There are also charged particles, W-bosons, for which it makes no sense to ask which of them is matter and which of them is antimatter. A W-boson may decay to a quark-antiquark pair so it's equally "far" from matter as it is from antimatter.
Neutrinos seem to be Majorana particles so far so they are identical to their antiparticles, too. However, the helicity (left-handed, right-handed) is correlated with the usual labels "pro-anti" which means that we can distinguish matter from antimatter, after all. There can also be right-handed neutrinos in which case the separation of neutrinos to "matter" and "antimatter" is exactly as possible (in principle) as it is for electrons and positrons.