  We study the boundedness of commutators of bi-parameter singular integrals
between mixed spaces
  $$
  [b,T]: L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}
  $$ in the off-diagonal situation $q_i,p_i\in(1,\infty)$ where we also allow
$q_i\not= p_i.$
  Boundedness is fully characterized for several arrangements of the
integrability exponents with some open problems presented.
