There are other neutral particles with antiparticles, such as the neutron and the $K^0$ meson. In those cases we have a microscopic theory that says those particles are made of quarks: for instance, the $K^0$ is made of a down quark and an anti-strange quark, while its antiparticle the $\bar K^0$ is made of a strange quark and an anti-down.
The neutrino is different from these because we have no evidence that it has any composite structure. While the neutrino doesn't have any electric charge, it does have a quantum number that appears to be conserved in the same way as electric charge: lepton number. We find in experiments that neutrinos are never created alone. A neutrino is always produced in conjunction with a positive lepton ($e$, $\mu$, or $\tau$), and an antineutrino is always produced in conjunction with a negative lepton.
There is another key property of neutrinos that's important when thinking about their antiparticles, which is their spin. Weak decays break mirror symmetry (or "parity symmetry"). If you have a beta-decay source that doesn't have any spin to it at all, and you measure the spins of the decay electrons that come out, you'll find that they are strongly polarized: beta-decay electrons prefer to be "left-handed", or traveling so that their south poles point forwards and their north poles point backwards. Beta-decay antielectrons, by contrast, prefer to be right-handed. The neutrinos follow the same rule: neutrinos have left-handed spins, and antineutrinos have right-handed spins.
If a neutrino had exactly zero mass, this polarization would be complete. However, we now have convincing evidence that at least two flavors of neutrino have finite mass. This means that it's possible, in theory, for an relativistic observer to "outrun" a left-handed neutrino, in which reference frame its north pole would be pointing along its momentum — that observer would consider it a right-handed neutrino. Would a right-handed neutrino act like an antineutrino? That would imply that the neutrino is actually its own antiparticle (an idea credited to Majorana). Would the right-handed neutrino simply refuse to participate in the weak interaction? That would make them good candidates for dark matter (though I think there is other evidence against this).
It's an open experimental question whether there is really a difference between neutrinos and antineutrinos, apart from their spin, and there are several active searches, e.g. for forbidden double-beta decays.
The lower mass limits of neutrinos is not 0eV. They have to have a mass, since we can observe neutrino oscillations. This is something the standard model did not get right. Looking at PDG (2014) the boundaries seem to be 0 < m < 2ev.
Best Answer
The detector that took that image--Super Kamiokande (super-K for short)--is a water Cerenkov device. It detects neutrinos by imaging the Cerenkov cone produced by the reaction products of the neutrinos. Mostly elastic scattering off of electrons: $$ \nu + e \to \nu + e \,,$$ but also quasi-elastic reactions like $$ \nu + n \to l + p \,,$$ where the neutron comes from the oxygen and $l$ means a charged lepton corresponding to the flavor of the neutrino (for energy reasons always an electron from solar neutrinos, but they also get muons from atmospheric and accelerator neutrinos---Super-K is the far detector for T2K).
Then you reconstruct the direction in which the lepton was moving (which is correlated with but not identical to the direction the neutrino was going). This indirect pointing method accounts for the very poor angular resolution of the image.