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 $\mathbb P_r(X ≥ x|H )$ for a one sided (right tailed) test. For a two sided (double tailed) test, this probability is doubled, meaning $ p = 2 \mathbb P_r(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 $ \mathbb P_r(X≥2|H)$ and not $2 \mathbb Pr(X≥2|H)$. 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 $ +\infty $ below the curve.
The question, therefore, is: Would it not be more mathematically accurate to compare $ \mathbb P_r(X≥2|H)$ to $\alpha/2$ $(=.025)$ instead of $2 \mathbb P_r(X≥2|H)$ 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 $2 \mathbb P_r(X≥x|H)$, then this $p$ is not the probability of $X$ being of greater magnitude than $x$, but rather this probability times $2$.