They're orthogonal, two dimensions that can be addressed independently.
Theism is to act as though one believes that there is one or more gods. Atheism is the negation of theism: to not have such a belief, and act accordingly. (Why action? Because that's where the decisions are made, and so where one actually has to commit to one belief or the other.)
Gnosticism is to have a belief that one *knows* something (in this case, that one's belief about gods is true). Agnosticism is the negation of gnosticism to not have that belief; i.e. to hold the position that one does not know.
Hence, one can be a gnostic theist (I know gods exist, and I act accordingly), or an agnostic theist (I don't know whether gods exist, but I act as though they do), or a gnostic atheist (I know gods do not exist, and I act accordingly), or an agnostic atheist (I don't know whether gods exist, but I act as though they do not).
Discussion (3)
While agnosticism != atheism, I'm not sure that this claim as structured is anything more than the tautology "!A : A :: !B : B" where the variables A is gnosticism and B is theism. I must admit that I am hoping for a more nuanced discussion about whether !A or !B are really true absolute negations.
Consider: Atheism is the negation of theism: to not have such a belief, and act accordingly.
Where would you put someone who refuses to decide whether there are gods or not? ... whose actions are neither dependent on there being gods or being no gods? ..who frankly doesn't care about the issue? Obviously, this is likely to be some kind of agnosticism, but it is neither theistic nor atheistic. Instead, it is the decision that one will honestly act in accordance with one's own full free will.
> "!A : A :: !B : B" where the variables A is gnosticism and B is theism.
Yes, that's the point; many people seem to think agnosticism is somewhere "between" theism and atheism on the same linear scale. That's incompatible with this claim.
> Where would you put someone who refuses to decide whether there are gods or not? ... whose actions are neither dependent on there being gods or being no gods? ..who frankly doesn't care about the issue?
"Cares about whether gods exist" is distinct from "acts as though gods do/do not exist". For example, one may not *care* whether there is a god who will answer any prayer to heal an illness, but when one chooses to go to the doctor for antibiotics, one is acting as though that god does not exist. Yes, this can be quibbled by the pedantic; but equally, the decision to be made can be tweaked so that it *does* hinge on whether the person believes in such a god.
Similar decisions people make every day reveal the extent to which they believe whatever "theism" represents. That's why I'm framing it in terms of actions, since that eliminates the variable of "cares/doesn't care".