Why don't antiparticles have antispin? Antiparticles are often said to particles moving backwards in time. However if they are moving backwards in time shouldn't they also "spin backwards", meaning they should have negative or antispin? But as far as I know antiparticles have the same spin as particles. Why is that?
 A: Why, yes, they do! 
Probably the most decisive way to show this involves "handedness" of neutrinos. I won't get too much into the details here, but neutrinos are nearly massless and travel at nearly the speed of light; and when something travels at the speed of light you can describe its spin as either "right-handed" (momentum and spin are in-line) or "left-handed" (momentum and spin are opposite). A crucial experiment was done I believe in the 60s where Europium absorbed an orbiting electron in reverse-beta-decay, growing into an unstable Strontium nucleus which then split apart emitting a photon; when the neutrino and photon are emitted back-to-back they must have opposite spins in order to preserve angular momentum, so we got to measure the neutrino's spin. This happened to prove that all neutrinos are left-handed, a big shock.
Because neutrinos are left-handed, antineutrinos are all right-handed. If there are right-handed neutrinos, or left-handed antineutrinos, then for some reason they don't seem to interact with normal matter anymore after that; we're not completely sure why, but neutrinos already were not very interacting particles to begin with, having no color charge for the strong nuclear force and no electromagnetic charge for the electromagnetic force.
However, we still say that an antineutrino is a spin-$\frac12$ particle. This is because its total spin is $S^2 = \hbar^2 \frac12 (\frac12 + 1)$, following the usual prescription of $\hbar^2 s (s + 1)$ for $s = \frac12$.This also means that $S_z = \pm \frac12 \hbar$ as anything larger than $\hbar/2$ or smaller than $-\hbar/2$ will not be able to stay within that bound for $S^2.$ In this way, we would not state that an antineutrino is a spin-$\frac{-1}{~~2}$ particle, that doesn't make any sense in the formula ($S^2$ would be negative?). It's just that the interactions that create an antineutrino will always generate the opposite spin from the corresponding time-reversed interaction where a neutrino is absorbed, and vice versa.
Prompted by @ACuriousMind below, I should also mention that after this experiment was done, results came from an awesome neutrino detector in Japan called "Super Kamiokande," which suggested strongly that neutrinos have "flavor oscillations" between electron-neutrinos, muon-neutrinos, and tau-neutrinos. The problem is that such flavor oscillations are only possible if the neutrino in fact does have mass, which breaks the above definition of "handedness:" helicity is not actually a Lorentz-invariant, since a reference frame which overtakes a left-handed neutrino sees it as a right-handed one, unless the particle is massless and therefore no reference frame can overtake it. So now the correct definition has to do with a more fundamental property called "chirality," which is the property of "this spin-$\frac12$ particle has a spinor wavefunction which flips by $-1$ under a 360-degree rotation; but which route around the complex plane does its spin follow in order to get from $1$ to $-1$ here?". See this link for a touchy-feely explanation of these ideas.
