Graduate Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense

[WWGD]

TL;DR

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive,

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense.
Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with $(M^{T})^{T}=M$, and just wanted to see if it made sense to talk about the " Alternative of the Alternative " returned the initial hypothesis.

 

[Dale]

TL;DR: The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive,

Since H0, HA aren't exhaustive

Why wouldn’t they be exhaustive?