When I take $X_1,...,X_n$ from $F(x)$. Then the ranks are $X_{(1)}<...<X_{(n)}$ and lets take the test-statistic, \begin{align} t_0(X_1,...,X_n) = \sum_{i=1}^n X_{(i)} \end{align}
If I look at $t(-X_1,...,-X_n)$ to see if $t(-X_1,...,-X_n)\overset{?}{=}-t(X_1,...,X_n)$, does the rank change?
For example is then $-X_1,...,-X_n$ correspond to $X_{(1)}<...<X_{(n)}$?
Sorry I am struggling to find the right way to ask this question.
EDIT
How about this test statistic, \begin{align} t_1(X_1,...,X_n) = \sum_{i=1}^n i\cdot X_{(i)} \end{align} Because I am wondering if the actual ranks change when I change their input values in $t_1(\cdot)$.
Is $t_1(-X_1,...,-X_n) = -t_1(X_1,...,X_n)$?
Your test statistic is simply $$t(X_1, \ldots , X_n) = \sum_{i=1}^n X_{(i)} = \sum_{i=1}^n X_{i} = n \overline{X}_n$$ If you flip the signs of each summand then you get $t(-X_1, \ldots , -X_n) = - n \overline{X}_n$.