  The $\Xi$-move is a local move generated by forbidden moves in virtual knot
theory. This move was introduced by Taniguchi and the second author, who showed
that it characterizes the odd writhe of virtual knots, which is a fundamental
invariant defined by Kauffman. In this paper, we extend this result by
classifying $2$-component virtual links up to $\Xi$-moves, using refinements of
the odd writhe and linking numbers.
