Hack for reversing page order after some point broken by recent versions of polyglossia

Question

In this question:

Reversing page order from a point onward

user @BrunoLeFloch offered a very useful hack for page reversal using the atbegshi package. Unfortunately, with recent versions of TeXLive, and when using both Hebrew and English, this triggers an error, and since the details are beyond my capabilities, I'd like to ask for help in understanding what's wrong. Here's a minimal(ish) example:

\documentclass{article}
\usepackage{atbegshi}
\makeatletter
\usepackage{polyglossia}
\setmainlanguage{english}
\setotherlanguage{french}
\newif\ifRP%
\newbox\RPbox%
\setbox\RPbox\vbox{\vskip1pt}%
\AtBeginShipout{%
  \ifRP
    \AtBeginShipoutDiscard%
    \global\setbox\RPbox\vbox{\unvbox\RPbox
      \box\AtBeginShipoutBox\kern\c@page sp}%
  \fi
}%
\renewcommand{\RPtrue}{% reverse page order
  \clearpage
  \ifRP\RPfalse\fi
  \global\let\ifRP\iftrue
}%
\renewcommand{\RPfalse}{% resume normal page order
  \clearpage
  \global\let\ifRP\iffalse
  \setbox\RPbox\vbox{\unvbox\RPbox
    \def\protect{\noexpand\protect\noexpand}%
    @whilesw\ifdim0pt=\lastskip\fi
      {\c@page\lastkern\unkern\shipout\lastbox}%
  }%
}%
\makeatother
\begin{document}
\RPtrue
\begin{french}
Fou
\newpage
\end{french}
Bar % Note: Originally, this question was posted without the extra text.
\RPfalse
\end{document}

The error message:

! Missing } inserted.
<inserted text> 
                }
l.48 \end{document}

I should mention this does work with TeXLive 2019.

---

edit: Here are the .aux files with TL 2021:

\relax 
\selectlanguage *{english}
\@writefile{toc}{\selectlanguage *{english}}
\@writefile{lof}{\selectlanguage *{english}}
\@writefile{lot}{\selectlanguage *{english}}
\bgroup 
\@writefile{toc}{\bgroup }
\@writefile{lof}{\bgroup }
\@writefile{lot}{\bgroup }
\selectlanguage *{french}
\@writefile{toc}{\selectlanguage *{french}}
\@writefile{lof}{\selectlanguage *{french}}
\@writefile{lot}{\selectlanguage *{french}}
\gdef \@abspage@last{1}

and with 2019:

\relax 
\protect \select@language {english}
\@writefile{toc}{\protect \select@language {english}}
\@writefile{lof}{\protect \select@language {english}}
\@writefile{lot}{\protect \select@language {english}}
\protect \select@language {french}
\@writefile{toc}{\protect \select@language {french}}
\@writefile{lof}{\protect \select@language {french}}
\@writefile{lot}{\protect \select@language {french}}

Answer 1

This is a partial answer but it is the best I can do in reasonable time.

The otherlanguage environment (used by \begin{french}...\end{french}) inserts on the page a "whatsit" (\write\@auxout{\bgroup} and \write\@auxout{\egroup}) at the start and at the end of the environment. When the page is actually shipped out to the pdf file, the two "whatsit" make TeX write \bgroup and \egroup to the aux file, to make some \selectlanguage settings be local inside the aux file. This serves to ensure the correct language rules are used for titles in the toc and so on.

Since we are reversing the order of pages, the \bgroup\selectlanguage{french}...\egroup becomes \egroup \bgroup\selectlanguage{french}..., which makes TeX complain about a bad nesting (\bgroup is the same as { and \egroup the same as }).

The simplest workaround is to use \selectlanguage{french} ... \selectlanguage{english} instead of the french environment, as I do below. However, in order for the language setting to be correct everywhere in the aux file, one needs to reselect the language often. I am not sure how to do that properly; one option seems to be to just put \selectlanguage{french} just before every section title, but probably this is not enough.

\documentclass{article}
\usepackage{lipsum}
\usepackage{atbegshi}
\makeatletter
\usepackage{polyglossia}
\setmainlanguage{english}
\setotherlanguage{french}
\newif\ifRP%
\newbox\RPbox%
\setbox\RPbox\vbox{\vskip1pt}%
\AtBeginShipout{%
  \ifRP
    \AtBeginShipoutDiscard%
    \global\setbox\RPbox\vbox{\unvbox\RPbox
      \box\AtBeginShipoutBox\kern\c@page sp}%
  \fi
}%
\renewcommand{\RPtrue}{% reverse page order
  \clearpage
  \ifRP\RPfalse\fi
  \global\let\ifRP\iftrue
}%
\renewcommand{\RPfalse}{% resume normal page order
  \clearpage
  \global\let\ifRP\iffalse
  \setbox\RPbox\vbox{\unvbox\RPbox
    \def\protect{\noexpand\protect\noexpand}%
    @whilesw\ifdim0pt=\lastskip\fi
      {\c@page\lastkern\unkern\shipout\lastbox}%
  }%
}%
\makeatother
\begin{document}
\section{Start in English}
\lipsum[1-10]
\RPtrue
\selectlanguage{french}
\section{Some French}
\lipsum[11-20]
Fou
\newpage
\selectlanguage{french} % needed just before any section/subsection and perhaps in figures etc.
\section{More French}
\lipsum[21-30]
\selectlanguage{english}
\section{English stuff}
\lipsum[31-40]
\RPfalse
\selectlanguage{english} % needed otherwise the last selectlanguage (after reversal) was french
\section{And finally some English}
\lipsum[41-50]
\end{document}

Other ways could be to only reverse pages with the same language, or on the contrary make sure that the language environments don't span any page break.

Answer 2

There is an internal (etoolbox-style) boolean (xpg@inbiditable) in polyglossia which you can use in your code to swap \bgroup and \egroup insertion to the aux (UPDATE: provide bool to account for comment below):

\documentclass{article}
\usepackage{atbegshi}
\makeatletter
\usepackage{polyglossia}
\setmainlanguage{english}
\setotherlanguage{french}
\newif\ifRP%
\newbox\RPbox%
\setbox\RPbox\vbox{\vskip1pt}%
\AtBeginShipout{%
    \ifRP
    \AtBeginShipoutDiscard%
    \global\setbox\RPbox\vbox{\unvbox\RPbox
        \box\AtBeginShipoutBox\kern\c@page sp}%
    \fi
}%
% This boolean is missing in older polyglossia versions
\providebool{xpg@inbiditable}
\renewcommand{\RPtrue}{% reverse page order
    \booltrue{xpg@inbiditable}
    \clearpage
    \ifRP\RPfalse\fi
    \global\let\ifRP\iftrue
}%
\renewcommand{\RPfalse}{% resume normal page order
    \boolfalse{xpg@inbiditable}
    \clearpage
    \global\let\ifRP\iffalse
    \setbox\RPbox\vbox{\unvbox\RPbox
        \def\protect{\noexpand\protect\noexpand}%
        @whilesw\ifdim0pt=\lastskip\fi
        {\c@page\lastkern\unkern\shipout\lastbox}%
    }%
}%
\makeatother
\begin{document}
\RPtrue
\begin{french}
Fou
\newpage
Bar
\end{french}
\RPfalse
\end{document}