The toffioli gate flips the target bit when both of the control bits are $\vert 1 \rangle $.
Would it be possible to instead have a gate which flips a target bit when both control bits are 'equal'?
Possible meanings of equal:
Control bits are identical: $\vert c1 \rangle = \alpha \vert 0 \rangle + \beta \vert 1 \rangle = \vert c2 \rangle$.
Control bits have the same sign/phase: doesn't flip if $\vert c1 \rangle = -\vert c2 \rangle$. Flips otherwise.
I am thinking about this in the context of qubits which may or may not have been marked by the oracle in Grover's algorithm if that helps.