Add a $\pm$ to the label in order to distinguish two different cases

Use \tag and manually step the equation counter.

Explanation: \begin{equation} steps up the counter; however \tag will cause the counter to be stepped down.

\documentclass[twocolumn]{article}
\usepackage{amsmath}

\begin{document}

We have a double equation
\begin{equation}\label{double}
f^{\pm}(x)=\dots
\tag{\theequation$^{\pm}$}
\stepcounter{equation}
\end{equation}
We also have a normal equation
\begin{equation}\label{single}
g(x)=\dots
\end{equation}
We can reference equation~\eqref{double} and equation~\eqref{single}.

How about~{\let\pm=+\eqref{double}} and {\let\pm=-\eqref{double}}?

\end{document}

Note: twocolumn is just to make a smaller picture.

An abstracted version that also works with hyperref.

\documentclass[twocolumn]{article}
\usepackage{amsmath}
\usepackage{hyperref}

\protected\def\hpm{\ensuremath{^\pm}}
\newcommand{\pmnumber}{\tag{\theequation\hpm}\stepcounter{equation}}
\newcommand{\eqrefp}[1]{{\let\pm=+\eqref{#1}}}
\newcommand{\eqrefm}[1]{{\let\pm=-\eqref{#1}}}

\begin{document}

We have a double equation
\begin{equation}\label{double}\pmnumber
f^{\pm}(x)=\dots
\end{equation}
We also have a normal equation
\begin{equation}\label{single}
g(x)=\dots
\end{equation}
We can reference equation~\eqref{double} and equation~\eqref{single}.

How about~\eqrefp{double} and \eqrefm{double}?

\end{document}

You can cheat by redefining \pm while referencing:

\documentclass{article}

\usepackage{amsmath}

\newcommand*\tagpm{\stepcounter{equation}\tag{\theequation\(\pm\)}}
\newcommand*\eqrefp[1]{{\def\pm{+}\eqref{#1}}}
\newcommand*\eqrefm[1]{{\def\pm{-}\eqref{#1}}}

\begin{document}

\begin{equation}\label{eqi}\tagpm
  f^\pm(x)=\ldots 
\end{equation}

\eqref{eqi}

\eqrefp{eqi}

\eqrefm{eqi}

\end{document}