  We study two modifications of the trapezoidal product cubature formulae,
approximating double integrals over the square domain $[a,b]^2=[a,b]\times
[a,b]$. Our modified cubature formulae use mixed type data: except evaluations
of the integrand on the points forming a uniform grid on $[a,b]^2$, they
involve two or four univariate integrals. An useful property of these cubature
formulae is that they are definite of order $(2,2)$, that is, they provide
one-sided approximation to the double integral for real-valued integrands from
the class $$ \mathcal{C}^{2,2}[a,b]=\{f(x,y)\,:\,\frac{\partial^4 f}{\partial
x^2\partial y^2}\ \text{continuous and does not change sign in}\ (a,b)^2\}. $$
For integrands from $\mathcal{C}^{2,2}[a,b]$ we prove monotonicity of the
remainders and derive a-posteriori error estimates.
