Neutrinos do not carry any electric charge. Neutrinos are electrically neutral. It is often argued that the “anti”-ness of a particle is related to the electrical charge. For example, electron has -1 electric charge and its anti-particle anti-electron which is also known as positron has +1 electric charge. Electron and positron are identical except for the electric charge. Extending this argument to neutrinos one could conclude that a neutrino has to be its own anti-particle because neutrino has no electric charge. So far, there is no experimental evidence that neutrinos are their own anti-particles.
Neutrino-less double beta decay has never been observed. There will be more sensitive searches in the future but the null result so far implies that neutrinos and anti-neutrinos are not the same.
“When an atom undergoes one type of beta decay, a neutron inside its nucleus spontaneously transforms into a proton, electron and antineutrino (the antimatter counterpart of the neutrino); in a type of inverse beta decay, the neutron absorbs a neutrino and morphs into a proton and electron. In neutrino-less double beta decay, both processes would happen in tandem: The antineutrino produced by the first type of decay would serve as the neutrino that enters into the second. Such a dual reaction can occur only if neutrinos and antineutrinos are one and the same particle, as the Italian physicist Ettore Majorana hypothesized in 1937.” *
* The article at the Quanta Magazine explains the latest experimental efforts to detect the neutrino-less double beta decay.