Proponents of neuro-linguistic programming (NLP) claim that you can tell if someone is lying based on how they move their eyes. This claim has also become popular in the public. For example, according to this web page, Bandler and Grinder, in their book on NLP say that right handed persons tend to look up to the left when they construct visual images such as when they are asked to imagine a purple buffalo. Since lying often involves construction of visual images (as opposed to things that have actually been witnessed), one should be able to tell if another person is lying by observing eye movements.
However, no experimental tests of this claim had been done until Richard Wiseman (who by the way have an excellent blog and youtube channel) and colleagues at the University of Edinburgh did a series of experiments on the topic. In the first part of the study (read the entire study here), they instructed participants to take a mobile phone, go into an office and then either put it in their pocket or put it in a specified drawer in the office.
Participants were then interviewed about their behavior. Everyone were told to say that they put the mobile phone in the drawer (which of course means that 50% would be lying). The interviewer did not know who was lying and who was telling the truth. Participants were filmed during the interview and subsequently two persons who were also blind to the different conditions estimated eye movements of the participants. If lying involves a specific pattern of eye movements, it should have been detected here. As it happened, there were no significant effects to be found anywhere. That is, there was no difference between the eye movements of participants who were lying and those who were telling the truth.
The study included two more experiments. In one subjects were told about the cues for detecting lies as taught by NLP proponents and were then asked to rate if the people from the first part of the study were lying or telling the truth. Their assessments were compared to those of subjects who had not learned about NLP rules. Again there was no difference in the accuracy of participants to determine whether someone was lying or telling the truth.
All in all, this study (I think), shows quite convincingly, that there is no truth to the idea that you can tell if someone is lying by observing their eye-movements.
Wiseman R, Watt C, ten Brinke L, Porter S, Couper SL, & Rankin C (2012). The eyes don't have it: lie detection and Neuro-Linguistic Programming. PloS one, 7 (7) PMID: 22808128