A p-value is "the probability for a given statistical model that, when the null hypothesis is true, the statistical summary would be the same as or of greater magnitude than the actual observed results." Source
Thus, the p-value is defined as Pr(X≥x|H) for a one sided (right tailed) test. For a two sided (double tailed) test, this probability is doubled, meaning p = 2 * Pr(X≥x|H) and then compared to alpha (e.g .05). However, how can this two sided p-value be interpreted as the probability as defined above? The real probability of getting a stat. summary of greater magnitude than the observed result (e.g. +2) is Pr(X≥2|H) and not 2 * Pr(X≥2|H). Yet, statisticians double the right-hand p-value. There is, of course, an error probability on the left side too (alpha/2), but this should be of no concern for us, as our observed value is positive and thus creates a probability only on its right side to +infinity below the curve.
The question, therefore, is: Would it not be more mathematically accurate to compare Pr(X≥2|H) to alpha/2 (=.025) instead of 2 * Pr(X≥2|H) to alpha (=.05) in a two sided test? The results would of course be the same in symmetric distributions, but not in asymmetric ones!
Furthermore, if we say that p for double tailed tests is indeed 2 * Pr(X≥x|H), then this p is not the probability of X being of greater magnitude than x, but rather this probability times 2.