The following example defines macros \arabicodd and \arabiceven that works similar to \arabic, but using odd and even numbers respectively.
As in the other answers of cmhughes and egreg, the list is made via package enumitem.
References work in the same way as for the "normal" enumerate environment, especially, the dot is not included in the reference. The appearance of the label and reference can easily be configured options label and/or ref. The counters macros \arabicodd and \arabiceven can be used as \arabic there, because they are registered using \AddEnumerateCounter.
\documentclass{article}
\usepackage{enumitem}
\makeatletter
\newcommand*{\arabicodd}[1]{%
\expandafter\@arabicodd\csname c@#1\endcsname
}
\newcommand*{\arabiceven}[1]{%
\expandafter\@arabiceven\csname c@#1\endcsname
}
\newcommand*{\@arabicodd}[1]{%
\@arabic{\numexpr(#1)*2-1\relax}%
}
\newcommand*{\@arabiceven}[1]{%
\@arabic{\numexpr(#1)*2\relax}%
}
\AddEnumerateCounter\arabicodd\@arabicodd{0}
\AddEnumerateCounter\arabiceven\@arabiceven{0}
\makeatother
\begin{document}
\begin{enumerate}
\item one
\item\label{normal:two}two
\item three
\end{enumerate}
Middle label of normal numbered list: \ref{normal:two}.
\begin{enumerate}[label=\arabicodd*.,ref=\arabicodd*]
\item one
\item\label{odd:two}two
\item three
\end{enumerate}
Middle label of odd-numbered list: \ref{odd:two}.
\begin{enumerate}[label=\arabiceven*.,ref=\arabiceven*]
\item one
\item\label{even:two}two
\item three
\end{enumerate}
Middle label of even-numbered list: \ref{even:two}.
\end{document}
