The package mathabx provides all arrows of this kind. The one you're looking for is \updownarrows.
\documentclass{article}
\usepackage{mathabx}
\begin{document}
$\upuparrows$ - $\downdownarrows$ - $\updownarrows$ - $\downuparrows$
\end{document}

For such questions, a good place to look at is the comprehensive LaTeX symbols list.
Note that mathabx redefines many symbols. An alternative is the following code, which produces updownarrows in a pure-latex fashion, but as you can see, the result is still not fully identical to amssymb's \upuparrows.
\documentclass{article}
\usepackage{amsmath,amssymb}
\newcommand{\updownarrows}{\mathbin\uparrow\hspace{-.5em}\downarrow}
\newcommand{\downuparrows}{\mathbin\downarrow\hspace{-.5em}\uparrow}
\begin{document}
$\upuparrows$ - $\downdownarrows$ - $\updownarrows$ - $\downuparrows$
\end{document}

A possible solution would be to redefine \upuparrows and \downdownarrows as well :
\documentclass{article}
\usepackage{amsmath,amssymb}
\newcommand{\updownarrows}{\mathbin\uparrow\hspace{-.3em}\downarrow}
\newcommand{\downuparrows}{\mathbin\downarrow\hspace{-.3em}\uparrow}
\renewcommand{\upuparrows}{\mathbin\uparrow\hspace{-.3em}\uparrow}
\renewcommand{\downdownarrows}{\mathbin\downarrow\hspace{-.3em}\downarrow}
\begin{document}
$\upuparrows$ - $\downdownarrows$ - $\updownarrows$ - $\downuparrows$
\end{document}

In comments, egreg proposed the following answer, for the same result (well, not technically the same), but with a cleaner code. It will probably give better spacing results if used within longer formulas :
\documentclass{article}
\usepackage{amsmath,amssymb}
\newcommand{\updownarrows}{\uparrow\mathrel{\mspace{-1mu}}\downarrow}
\newcommand{\downuparrows}{\downarrow\mathrel{\mspace{-1mu}}\uparrow}
\renewcommand{\upuparrows}{\uparrowuparrow}
\renewcommand{\downdownarrows}{\downarrow\downarrow}
\begin{document}
$\upuparrows$ - $\downdownarrows$ - $\updownarrows$ - $\downuparrows$
\end{document}

\updownarrows– Ipsen Dec 9 '12 at 13:29