What is the correct term for the "polarity" of matter (matter vs. antimatter)? Are fractional polarities allowed? What is the correct term for the "polarity" of matter (matter vs. antimatter)?
Are neutral polarities allowed? (1,0,-1)
Are fractional polarities allowed?
 A: Good question! There actually isn't a term for this that I know of. The most common use of such a term would be to classify a particle, for example "the 'polarity' of the electron is matter-polarity," but in that case most physicists would just say "the electron is a matter particle."
There is a mathematical operator called the charge conjugation operator, which converts the mathematical description of a matter particle into its corresponding antiparticle. This can sort of be used to distinguish matter and antimatter, though it doesn't tell you which particle is matter and which particle is antimatter (because that is just a matter of which one happens to be more common).
There are two known elementary particles, the photon and the $Z^0$ boson, whose mathematical description is not changed by the application of the charge conjugation operator. In other words, the photon and $Z^0$ are their own antiparticles. This corresponds to what you're calling "neutral polarity." We know of other particles which are their own antiparticle because they are a combination of a more fundamental particle and its antiparticle; for example, the $\mathrm{J/\psi}$ meson is made of a charm quark and an anticharm quark, and so switching matter to antimatter and vice versa gives you the $\mathrm{J/\psi}$ right back.
Given the definition of charge conjugation, I don't think fractional "polarities" are really a sensible concept. After all, it should make sense that applying charge conjugation twice is supposed to convert a particle back into its original self, perhaps up to a sign: $|\mathcal{C}^2| = 1$, and I think a fractional "polarity" by definition would imply that $|\mathcal{C}^2| \neq 1$. What seems a little less strange is a "complex polarity," where $|\mathcal{C}^n| = 1$ for some $n > 2$; this would correspond to a particle (or rather, its mathematical description) which would have to be charge-conjugated multiple times to get back to its original self. So instead of just a particle and antiparticle, you'd have the particle, conjugate particle 1, conjugate particle 2, etc., and in order to get them to annihilate you would have to bring one of each of these different particles all together at the same place at the same time. Nothing like that is known to exist, and I wouldn't be surprised if there is some theoretical reason to believe it can't exist. (It does bear some similarities to the color charge associated with the strong force, though.)
A: There is no single polarity or more correctly conserved quantum number. It depends on the nonzero quantum numbers the particle has. For example for electrons and positrons it would be both the electrical charge and the electron flavor. There are particles that are there own antiparticle such as the photon. So for a photon all of these numbers are 0 which is why it is its own antiparticle. 
Fractional numbers are allowed,  for example quarks have electrical charges of $\pm\frac{1}{3}$ and $\pm\frac{2}{3}$.
Edit: I would add that the stable atoms in our universe are made from electrons and up & down quarks.  So these are called matter particles whereas positrons and anti-up & anti-down quarks are called antimatter particles.  In addition any higher mass unstable particles that decays predominantly into these "matter" particles area also call matter particles and if they decay into antimatter particles.
