“Logically symmetrical”

Suppose we say:

“The theory is supported because there is no evidence against it.”

And someone responds:

“The theory is unsupported because there is no evidence for it.”

These statements are logically symmetrical in the sense that they make opposite claims from the same absence of evidence.

But absence of evidence does not automatically establish either conclusion.

You need to ask what evidence should reasonably exist if the proposition were true.