We give several new characterizations of the dual of the dyadic
Hardy space $H^{1,d}(\mathbb{T}^2)$, the so-called dyadic BMO space
in two variables and denoted ${\mathrm{BMO}}^{\mathit d}_{prod}}$.
These include characterizations in terms of Haar multipliers,
in terms of the ``symmetrised paraproduct'' $\Lambda_b$, in terms
of the rectangular BMO norms of the iterated ``sweeps'', and in
terms of nested commutators with dyadic martingale transforms.
We further explore the connection between ${\mathrm{BMO}}^{\mathit d}_{prod}}$
and John-Nirenberg type inequalities, and study a scale of rectangular
BMO spaces.