  We prove compactness results and characterizations for the bi-commutator
$[T_1,[b, T_2]]$ of a symbol $b$ and two non-degenerate Calder\'on-Zygmund
singular integral operators $T_1, T_2$. Our strategy for proving sufficient
conditions for compactness is to first establish them in the mixed-norm
$L^{p_1}L^{p_2}\to L^{q_1}L^{q_2}$ off-diagonal case with $p_i < q_i$, and then
extend these to other exponents, including the diagonal $p_i = q_i$, with a new
extrapolation argument. In particular, the natural product $\operatorname{VMO}$
condition is obtained as a sufficient condition in the diagonal. A full
characterization is obtained, both in terms of a vanishing mean oscillation
type condition and in terms of the approximability of the symbol, whenever the
inequality $p_i \le q_i$ is strict for at least one index. The extrapolation
scheme for proving sufficiency requires us to prove new approximation results
in relevant bi-parameter function spaces that are of independent interest. The
necessity results are obtained by carefully combining recent rectangular
approximate weak factorization methods with a classical idea of Uchiyama.
