My situation is as follows: I want, through a Monte-Carlo study, to compare $p$-values of two different tests for statistical significance of an estimated parameter (null is "no effect - parameter is zero", and the implied alternative is "parameter is not zero"). Test A is the standard "independent two-sample t-test for equality of means", with equal variances under the null.
Test B I have constructed myself. Here, the null distribution used is an asymmetric generic discrete distribution. But I have found the following comment in Rohatgi & Saleh (2001, 2nd ed, p. 462)
"If the distribution is not symmetric, the $p$-value is not well defined in the two-sided case, although many authors recommend doubling the one-sided $p$-value".
The authors do not discuss this further, nor do they comment on the "many authors suggestion" to double the one-sided $p$-value. (This creates the question "double the $p$-value of which side? And why this side and not the other?)
I was not able to find any other comment, opinion or result on this whole matter. I understand that with an asymmetric distribution although we can consider an interval symmetric around the null hypothesis as regards the value of the parameter, we will not have the second usual symmetry, that of probability mass allocation. But I do not understand why this makes the $p$-value "not-well defined". Personally, by using an interval symmetric around the null hypothesis for the values of the estimator I see no definitional problem in saying "the probability that the null distribution will produce values equal to the boundaries of, or outside this interval is XX". The fact that the probability mass on the one side will be different than the probability mass on the other side, does not appear to cause troubles, at least for my purposes. But it is rather more probable than not that Rohatgi & Saleh know something that I don't.
So this is my question: In what sense the $p$-value is (or can be) "not well defined" in the case of a two-sided test when the null distribution is not symmetric?
A perhaps important note: I approach the matter more in a Fisherian spirit, I am not trying to obtain a strict decision rule in the Neyman-Pearson sense. I leave it up to the user of the test to use the $p$-value information alongside any other information to make inferences.