Controlling for response bias in a forced choice memory task

I want to present participants a pair of items $a$ and $b$, each with an association $x$ or $y$. Later on participants will presented with the items singularly, and asked 'what was the association?'. I'm interested in differences in their memory for the association, between levels of the association.

As I've specified it there is no way to separate differences in memory performance between levels from response bias, because being incorrect on association $x$ means that another response for association $y$ is recorded. Signal detection methods seem to give me an overall measure of discriminability between the two and bias, but can't tell me if the bias is due to genuinely improved memory for $x$, or simply a predisposition to respond $x$.

What is the best way to deal with this?

Initial thoughts: I'm considering including previously unseen elements into the memory task and getting a measure of response bias that way.

Here is some random example data to help with thinking:

choice X | X    choice Y | X   choice Y | Y   choice X | Y
18               18         18              18
24               12         18              18
30                6         14              22
29                7         14              22
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