  We develop both bilinear theory and commutator estimates in the context of
entangled dilations, specifically Zygmund dilations $(x_1, x_2, x_3) \mapsto
(\delta_1 x_1, \delta_2 x_2, \delta_1 \delta_2 x_3)$ in $\mathbb{R}^3$. We
construct bilinear versions of recent dyadic multiresolution methods for
Zygmund dilations and apply them to prove a paraproduct free $T1$ theorem for
bilinear singular integrals invariant under Zygmund dilations. Independently,
we prove linear commutator estimates even when the underlying singular
integrals do not satisfy weighted estimates with Zygmund weights. This requires
new paraproduct estimates.
