For slightly improved spacing displayed and inline, you can use \ooalign
. The code is
\newcommand{\twprod}{\mathbin{%
\ooalign{\raise1.15ex\hbox{$\scriptstyle\sim$}\cr\hidewidth$\times$\hidewidth\cr}%
}}

If you will ever use the symbol in a subscript or superscript, \mathchoice
allows for greatly improved spacing in scriptstyle. The height of the \sim
is easily adjustable by changing the 1.15ex
for display and inline, .85ex
for script and (though I can't imagine using it) .65ex
for scriptscript.
\documentclass{article}
\newcommand{\twprod}{\mathbin{\mathchoice%
{\ooalign{\raise1.15ex\hbox{$\scriptstyle\sim$}\cr\hidewidth$\times$\hidewidth\cr}}%
{\ooalign{\raise1.15ex\hbox{$\scriptstyle\sim$}\cr\hidewidth$\times$\hidewidth\cr}}%
{\ooalign{\raise.85ex\hbox{$\scriptscriptstyle\sim$}\cr\hidewidth$\scriptstyle\times$\hidewidth\cr}}%
{\ooalign{\raise.65ex\hbox{$\scriptscriptstyle\sim$}\cr\hidewidth$\scriptscriptstyle\times$\hidewidth\cr}}%
}}
\begin{document}
$S^2\twprod S^2\quad F_{S^2\twprod S^2}$
\end{document}