Change equation numbering between left and right

Question

Normally LaTeX puts equation numbers at the righthand margin. I know that this can be changed by using the leqno class option to put them at the lefthand margin. What I would like is to have local control over which margin the numbers are adjacent to. Trying somewhat optimistically

\documentclass{article}
\begin{document}
\begin{equation}
a + b = c          % numbered at right
\end{equation}

\bgroup
\leqno            % ERROR can't use \leqno in vertical mode
\begin{equation}
a + b = c
\end{equation}
\egroup

\begin{equation} \leqno  % ERROR can't use \eqno in math mode
a + b = c
\end{equation}

\end{document}

As you can see my attempts failed. Any suggestions?

EDIT The above MWE didn't tell the whole story. I should have given the initial code more appropriately as:

\documentclass{memoir}
\usepackage{memsty}    % pages of code
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{newpxtext}
\usepackage{newpxmath}
\begin{document}
 ...

Both answers solved my problem, and it was a close-run thing between them, as initially presented but ...

I will look into the problems generated by the various ...math... packages and hope to resolve these. The memsty package I will deal with. You may find, however, another question on the topic at a later date, but I hope not.

Answer 1

\leqno is a primitive and not used in latex. The option in latex redefines \@eqnnum:

\documentclass{article}
\makeatletter
\newcommand\useleqno{\renewcommand\@eqnnum{\hb@xt@.01\p@{}%
                      \rlap{\normalfont\normalcolor
                        \hskip -\displaywidth(\theequation)}}}
\begin{document}
\begin{equation}
a + b = c          % numbered at right
\end{equation}

\bgroup
\useleqno
\begin{equation}
a + b = c
\end{equation}
\egroup

\begin{equation} 
a + b = c
\end{equation}

\end{document}

Answer 2

The commands only work in displaymath mode. Try:

$$ E = m c^2 \eqno (1) $$
$$ E = m c^2 \leqno (2) $$

Your examples cannot work as the definion of \endequation already uses \eqno:

% from latex.ltx
\def\equation{$$\refstepcounter{equation}}
\def\endequation{\eqno \hbox{\@eqnnum}$$\@ignoretrue}

So you can define your own environment (full MWE):

\documentclass{article}
% The definitions:
\makeatletter
\let\endeqno@equation\endequation
\def\endleqno@equation{\leqno \hbox{\@eqnnum}$$\@ignoretrue}
%
\def\lefteqnnum{\let\endequation\endleqno@equation}
\def\righteqnnum{\let\endequation\endeqno@equation}
\makeatother

\begin{document}

\begin{equation} E=mc^2 \end{equation}

\lefteqnnum
\begin{equation} E=mc^2 \end{equation}

\righteqnnum
\begin{equation} E=mc^2 \end{equation}

\end{document}