Symbol for a "twisted product"?

Detexify didn't help, so maybe you really have to do it yourself:

\newcommand\simtimes{\mathbin{%
    \stackrel{\sim}{\smash{\times}\rule{0pt}{0.9ex}}%
    }}

Adjust the 0.9ex to your needs.

The parameters 0.7 and 0.3 might be needed adjustment if used with different fonts.

\documentclass{article}
\usepackage{amsmath}
\usepackage{graphicx}

% a general purpose macro
\newcommand{\xmathpalette}[2]{\mathchoice
  {#1\displaystyle\textfont{#2}}%
  {#1\textstyle\textfont{#2}}%
  {#1\scriptstyle\scriptfont{#2}}%
  {#1\scriptscriptstyle\scriptscriptfont{#2}}%
}

\makeatletter
\newcommand{\twistedproduct}{%
  \mathbin{\xmathpalette\twisted@product\relax}%
}

\newcommand{\twisted@product}[3]{%
  \vbox{%
    \ialign{\hfil##\hfil\cr
      \scalebox{0.7}{$\m@th#1\sim$}\cr
      \noalign{\nointerlineskip\kern-0.3\fontdimen5 #2 2}
      $\m@th#1\times$\cr
    }%
  }%
}
\makeatother

\begin{document}

$S^2\twistedproduct S^2$ $X_{\twistedproduct_{\twistedproduct}}$

\end{document}

The macro \xmathpalette is an extension of \mathpalette that makes also the font in the current style (\textfont, \scriptfont or \scriptscriptfont). In this case we use it to access the x-height of the symbol font for the current style.

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}