Is there a "p-value" for the setting where the assumption is on the alternative? Consider testing the null hypothesis $H_0: X \sim f_0 $ versus $H_1: X \sim f_1 $, where $X$ is  non-negative. Assume also that the elements coming from $f_1$ are more concentrated in higher values while elements coming from $f_0$ are closer to zero.
The convention is to have an assumption on $f_0$, and then compute the p-value for a realization $x$ as $Pr(X>x|f_0)$.
My question is, does $Pr(X>x|f_1)$ hold any useful meaning? That's the probability of getting a value at least as extreme as $x$ if the data is coming from $f_1$. If this value is low, then it still seems to logically indicate that the element is from $f_1$, right? However, it's not technically a p-value.

Yes, it is meaningful
Under the stated hypotheses, your p-value function is:
$$p_0(x) = \mathbb{P}(X \geqslant x | f_0).$$
If you were to reverse the hypotheses, then your p-value function would be:
$$p_1(x) = \mathbb{P}(X \leqslant x | f_1).$$
So, as you can see, you have $\mathbb{P}(X>x|f_1) = 1-p_1(x)$ and so the probability of interest to you is one minus the p-value that would be used under the swapped test.  Since the p-value is a function of this probability, you can reasonably say that that probability fully defines the evidence in the swapped test.
