<p style='text-indent:20px;'>Marcinkiewicz integral operators on product domains defined by translates determined by twisted surfaces are introduced. Maximal functions along twisted surfaces are also introduced. Conditions on the underlined surfaces implying that the corresponding Marcinkiewicz integral operators map <inline-formula><tex-math id="M1">\begin{document}$ L^{p}\rightarrow L^{p} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$

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... &lt;\infty $\end{document}</tex-math></inline-formula> are obtained. A general method involving lacunary families of multi-indices is developed.</p>