  We study a trilinear singular integral form acting on two-dimensional
functions and possessing invariances under arbitrary matrix dilations and
linear modulations. One part of the motivation for introducing it lies in its
large symmetry groups acting on the Fourier side. Another part of the
motivation is that this form stands between the bilinear Hilbert transforms and
the first Calder\'on commutator, in the sense that it can be reduced to a
superposition of the former, while it also successfully encodes the latter. As
the main result we determine the exact range of exponents in which the ${L}^p$
estimates hold for the considered form.
