Right-justifying a comment for an equation that uses leqno

I'm unable to get the formatting correct when attempting to add a comment to an equation that uses leqno. Ideally, I'd like the equation to remain centered regardless of whether or not it is numbered. Any suggestions would be greatly appreciated.

\documentclass[12pt, leqno]{book}
\usepackage{amsmath}
\begin{document}
\noindent The equation is centered and does not contain a comment. Everything looks fine here.
$$(a+b)^2=a^2+2ab+b^2$$
Adding a comment using align no longer centers the equation and the comment is not right justified.
\begin{align}
(a+b)^2&=a^2+2ab+b^2 && \text{Right justify me!}
\end{align}
Commenting on an unnumbered equation causes equation to become uncentered and comment is on the wrong side.
\begin{equation*}
(a+b)^2=a^2+2ab+b^2 \tag*{(Wrong side) }
\end{equation*}
\end{document}

You can do what you want with flalign and \llap:

\documentclass[12pt, leqno]{book}
\usepackage{amsmath}

\begin{document}

\noindent The equation is centred and does not contain a comment. Everything looks fine here.
$$(a+b)^2=a^2+2ab+b^2$$

With \verb+flalign+ and \verb+\llap+, it remains centred:
\begin{flalign}
& & (a+b)^2&=a^2+2ab+b^2 && \llap{Right justify me!}
\end{flalign}

\end{document}

Possibly not very satisfying for you, but flalign does the job:

\documentclass[12pt, leqno]{book}
\usepackage{amsmath}
\begin{document}
\noindent The equation is centered and does not contain a comment. Everything looks fine here.
$$(a+b)^2=a^2+2ab+b^2$$
Adding a comment using align no longer centers the equation and the comment is not right justified.
\begin{flalign}
&&(a+b)^2=a^2+2ab+b^2
&&\text{Right justify me!}
\end{flalign}
\end{document}

You may want to insert by hand some space between the first two &. On the other hand, I don't think that flushing the comment to the right really helps the understanding of the equation.

As described in this question, you could switch between leqno und reqno and then use the \tag*{•} macro.

\documentclass[12pt]{book}
\usepackage[leqno]{amsmath}

\makeatletter
\newcommand{\leqnomode}{\tagsleft@true}
\newcommand{\reqnomode}{\tagsleft@false}
\makeatother

\begin{document}
$$(a+b)^2=a^2+2ab+b^2$$

\reqnomode
\begin{align*}
(a+b)^2=a^2+2ab+b^2 \tag*{Right side}
\end{align*}
\leqnomode

$$(a+b)^2=a^2+2ab+b^2$$

\end{document}

Output: