In a two sided test, assume the test statistic has been chosen to be $T(X)$ and the distribution of $T(X)$ under null hypothesis is also known to be $F$. Let the significance level be $\alpha$.
I can come up with two different ways to determine the rejection region:
$\{|T(x) - \mu| > c\}$. $\mu$ is the mean of the distribution $F$ of $T(X)$under null, and $c$ is determined by solving $$\inf_{c \geq 0} c$$ subject to $$P_{T(X) \sim F} (|T(X) - \mu| > c) \leq \alpha.$$ So the rejection region is symmetric around $\mu$.
$\{T(x) > c_1\} \cup \{T(x) < c_2\}$. $c_1$ and $c_2$ are determined by solving $$\inf_{c_1 \in \mathbb R} c_1$$ subject to $$P_{T(X) \sim F} (T(X) > c_1) \leq \alpha/2$$ and $$\sup_{c_2 \in \mathbb R} c_2$$ subject to $$P_{T(X) \sim F} (T(X) < c_2) \leq \alpha/2.$$ So the rejection region evenly split $\alpha$ to both sides.
Questions:
Am I correct that those two methods will agree when the null distribution $F$of $T(X)$ is symmetric around its mean $\mu$, and may not agree when $F$ isn't symmetric around $\mu$?
I was wondering what advantage and disadvantages these two methods have? Which one is recommended and when?
Are both methods used in some textbooks? If yes, references?
What are some other methods for two-sided tests? For example, can we generalized the second method by splitting $\alpha$ arbitrarily unevenly to the two sides?
Consider the relation between rejection region in testing and confidence interval. Are the above discussions also apply to confidence intervals?
Thanks and regards!