  We study the bi-commutators $[T_1, [b, T_2]]$ of pointwise multiplication and
Calder\'on-Zygmund operators, and characterize their $L^{p_1}L^{p_2} \to
L^{q_1}L^{q_2}$ boundedness for several off-diagonal regimes of the mixed-norm
integrability exponents $(p_1,p_2)\neq(q_1,q_2)$. The strategy is based on a
bi-parameter version of the recent approximate weak factorization method.
