Quoting from Haas 1984, Experiment 3 which is the one examining individual differences.
Sixty-two undergraduates (15 men and 47 women) participated in Experiment 3.
Power was very low, hence the correlation cannot be reliably estimated.
He also combined this experiment with a manipulation of self-focused attention (showing or hiding a tape recorder during the task).
The effect measured public self-consciousness (using a median-split which wastes information) was not significant:
Chi.square (1, N = 62) = 2.85, p < .10, and the interaction between manipulated 2 self-focus and public self-consciousness, Chi.square (1, N = 62) = 3.09, p < .08, on E direction was not significant either.
The author wrote:
Apparently the presence of the tape recorder in the situation overwhelmed the individual difference variable in the high self-focus condition, so that no difference between high and low public self-conscious subjects could be found. When the self-focus inducing effects of the tape recorder were removed from the situation, however, the relationship between the personality variable and the direction the E was drawn emerged.
He then did post-hoc analyses (known to increase false positives, but it is hard to appropriately correct for this)
Looking at the results in a slightly different way, correlations were computed between each of the Self-Consciousness Scale subscale scores and perspective taking. For all subjects com- bined a significant correlation of .36 (p < .01) was found between public self-consciousness and the likelihood of drawing the E from an external perspective. The correlation between the direction the E was drawn and private self- consciousness was not significant (r = .08). A marginally significant correlation was found between E direction and social anxiety (r = .21, p = .10), but this correlation declined to .12 (ns) when the effect of public self-consciousness was removed using a partial correlation.
The 95% confidence interval of a .31 correlation with 62 subjects is [0.07; 0.52].
All told, if this is all Wiseman is relying on (I didn't watch the video, I trusted @ndwaldner), then there is quite the jump from a just-significant correlation between E direction and public self consciousness to Q direction and being "more of an intravert and not very good at lying".
Obviously, the self consciousness scale relied on self-report. Personality psychologists usually don't worry about that, but when you're making inferences about lying you probably should. Of course it'd easy to change the E/Q direction consciously if you have a hunch of what is being assessed.
There has been a number of conceptual replications of this paper, which I didn't read.