signals for values of L and E that are not consistent with the Lorentz-invariant explanation. For example, KamLAND and SNO observations require a mass-squared difference to be consistent with the Lorentz-invariant phase proportional to L/E. Similarly, Super-Kamiokande, K2K, and MINOS observations of atmospheric-neutrino oscillations require a mass-squared difference . Any neutrino-oscillation experiment must be consistent with either of these two mass-squared differences for Lorentz invariance to hold. To date, this is the only class of signal for which there is positive evidence. The LSND experiment observed oscillations leading to a mass-squared difference that is inconsistent with results from solar- and atmospheric-neutrino observations. The oscillation phase requires . This anomaly can be understood in the presence of Lorentz violation. Periodic variations Laboratory experiments follow complicated trajectories as the Earth rotates on its axis and revolves around the Sun. Since the fixed SME background fields are coupled with the particle fields, periodic variations associated with these motions would be one of the signatures of Lorentz violation. There are two categories of periodic variations: Sidereal variations: As the Earth rotates, the source and detector for any neutrino experiment will rotate along with it at a sidereal frequency of . Since the 3-momentum of the neutrino beam is coupled to the SME background fields, this can lead to sidereal variations in the observed oscillation probability data. Sidereal variations are among the most commonly sought signals in Lorentz tests in other sectors of the SME. Annual variations: Variations with a period of one year can arise due to the motion of the Earth around the Sun. The mechanism is the same as for sidereal variations, arising because the particle fields couple to the fixed SME background fields. These effects, however, are challenging to resolve because they require the experiment to provide data for a comparable length of time. There are also boost effects that arise because the earth moves around the Sun at more than 30 kilometers per second. However, this is one ten thousandth of the speed of light, and means the boost effects are suppressed by four orders of magnitude relative to purely rotational effects. Compass asymmetries The breaking of rotation invariance can also lead to time-independent signals arising in the form of directional asymmetries at the location of the detector. This type of signal can cause differences in observed neutrino properties for neutrinos originating from different directions. Neutrino-antineutrino mixing Some of the