You can define different page styles for odd and even pages with fancyhdr.
\documentclass[latin,english]{book}
\usepackage{babel}
\usepackage{Libertinus}
\usepackage{fancyhdr}
\usepackage{microtype}
\pagestyle{fancy}
\fancyhead[L]{\textsc{Cicero}}
\fancyhead[RE]{\textit{On the Ends of Good and Evil Things}}
\fancyhead[RO]{\textit{\foreignlanguage{latin}{De Finibus Bonorum et Malorum}}}
\fancyfoot[CE]{\arabic{page}}
\fancyfoot[CO]{\roman{page}}
\title{\textsc{De Finibus Bonorum et Malorum}}
\author{Cicero}
\date{}
\begin{document}
\maketitle
But I must explain to you how all this mistaken idea of reprobating pleasure
and extolling pain arose. To do so, I will give you a complete account of the
system, and expound the actual teachings of the great explorer of the truth,
the master-builder of human happiness. No one rejects, dislikes or avoids
pleasure itself, because it is pleasure, but because those who do not know
how to pursue pleasure rationally encounter consequences that are extremely
painful. Nor again is there anyone \textbf{who loves or pursues or desires to obtain
pain of itself, because it is pain,} but occasionally circumstances occur in which
toil and pain can procure him some great pleasure. To take a trivial example,
which of us ever undertakes laborious physical exercise, except to obtain
some advantage from it? But who has any right to find fault with a man who
chooses to enjoy a pleasure that has no annoying consequences, or one who
avoids a pain that produces no resultant pleasure?] On the other hand, we
denounce with righteous indignation and dislike men who are so beguiled and
demoralized by the charms of pleasure of the moment, so blinded by desire,
that they cannot foresee the pain and trouble that are bound to ensue; and
equal blame belongs to those who fail in their duty through weakness of will,
which is the same as saying through shrinking from toil and pain. These cases
are perfectly simple and easy to distinguish. In a free hour, when our power of
choice is untrammeled and when nothing prevents our being able to do what we
like best, every pleasure is to be welcomed and every pain avoided. But in
certain circumstances and owing to the claims of duty or the obligations of
business it will frequently occur that pleasures have to be repudiated and
annoyances accepted. The wise man therefore always holds in these matters
to this principle of selection: he rejects pleasures to secure other greater
pleasures, or else he endures pains to avoid worse pains.
\clearpage
\begin{otherlanguage}{latin}
Sed ut perspiciatis, unde omnis iste natus error sit voluptatem accusantium
doloremque laudantium, totam rem aperiam eaque ipsa, quae ab illo inventore
veritatis et quasi architecto beatae vitae dicta sunt, explicabo. Nemo enim
ipsam voluptatem, quia voluptas sit, aspernatur aut odit aut fugit, sed quia
consequuntur magni dolores eos, qui ratione voluptatem sequi nesciunt,
neque porro quisquam est, \textbf{qui do lorem ipsum, quia dolor sit amet}
consectetur adipisci[ng] velit, sed quia non numquam [do] eius modi tempora
inci[di]dunt, ut labore et dolore magnam aliquam quaerat voluptatem. Ut enim
ad minima veniam, quis nostrum[d] exercitationem ullam corporis suscipit
laboriosam, nisi ut aliquid ex ea commodi consequatur? [D]Quis autem vel eum
iure reprehenderit, qui in ea voluptate velit esse, quam nihil molestiae
consequatur, vel illum, qui dolorem eum fugiat, quo voluptas nulla pariatur?
At vero eos et accusamus et iusto odio dignissimos ducimus, qui blanditiis
praesentium voluptatum deleniti atque corrupti, quos dolores et quas molestias
excepturi sint, obcaecati cupiditate non provident, similique sunt in culpa, qui
officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum
quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum
soluta nobis est eligendi optio, cumque nihil impedit, quo minus id, quod
maxime placeat, facere possimus, omnis voluptas assumenda est, omnis dolor
repellendus. Temporibus autem quibusdam et aut officiis debitis aut rerum
necessitatibus saepe eveniet, ut et voluptates repudiandae sint et molestiae
non recusandae. Itaque earum rerum hic tenetur a sapiente delectus, ut aut
reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus
asperiores repellat.
\end{otherlanguage}
\end{document}
Since this will go wrong as soon as the page-breaking algorithm doesn’t do what you expect, you might want to typeset each page inside an environment that prevents page breaking, such as minipage.
Within a \begin{otherlanguage} environment, babel will translate captions, such as “Chapter,” for you.
If you cannot strictly alternate, you can define your own englishpage and latinpage environments that change the language, and also override \fancyhead and fancyfoot.
\documentclass[latin,english]{book}
\usepackage{babel}
\usepackage{Libertinus}
\usepackage{fancyhdr}
\usepackage{microtype}
\pagestyle{fancy}
\fancyhead[L]{\textsc{Cicero}}
\fancyhead[R]{\textit{On the Ends of Good and Evil Things}}
\fancyfoot[C]{\arabic{page}}
\title{\textsc{De Finibus Bonorum et Malorum}}
\author{Cicero}
\date{}
\newenvironment{englishpage}%
{\selectlanguage{english}%
\fancyhead[R]{\textit{On the Ends of Good and Evil Things}}%
\fancyfoot[C]{\arabic{page}}}
{\clearpage}
\newenvironment{latinpage}%
{\selectlanguage{latin}%
\fancyhead[R]{\textit{De Finibus Bonorum et Malorum}}%
\fancyfoot[C]{\Roman{page}}}
{\clearpage}
\begin{document}
\maketitle
\begin{englishpage}
But I must explain to you how all this mistaken idea of reprobating pleasure
and extolling pain arose. To do so, I will give you a complete account of the
system, and expound the actual teachings of the great explorer of the truth,
the master-builder of human happiness. No one rejects, dislikes or avoids
pleasure itself, because it is pleasure, but because those who do not know
how to pursue pleasure rationally encounter consequences that are extremely
painful. Nor again is there anyone \textbf{who loves or pursues or desires to obtain
pain of itself, because it is pain,} but occasionally circumstances occur in which
toil and pain can procure him some great pleasure. To take a trivial example,
which of us ever undertakes laborious physical exercise, except to obtain
some advantage from it? But who has any right to find fault with a man who
chooses to enjoy a pleasure that has no annoying consequences, or one who
avoids a pain that produces no resultant pleasure?] On the other hand, we
denounce with righteous indignation and dislike men who are so beguiled and
demoralized by the charms of pleasure of the moment, so blinded by desire,
that they cannot foresee the pain and trouble that are bound to ensue; and
equal blame belongs to those who fail in their duty through weakness of will,
which is the same as saying through shrinking from toil and pain. These cases
are perfectly simple and easy to distinguish. In a free hour, when our power of
choice is untrammeled and when nothing prevents our being able to do what we
like best, every pleasure is to be welcomed and every pain avoided. But in
certain circumstances and owing to the claims of duty or the obligations of
business it will frequently occur that pleasures have to be repudiated and
annoyances accepted. The wise man therefore always holds in these matters
to this principle of selection: he rejects pleasures to secure other greater
pleasures, or else he endures pains to avoid worse pains.
\end{englishpage}
\begin{latinpage}
Sed ut perspiciatis, unde omnis iste natus error sit voluptatem accusantium
doloremque laudantium, totam rem aperiam eaque ipsa, quae ab illo inventore
veritatis et quasi architecto beatae vitae dicta sunt, explicabo. Nemo enim
ipsam voluptatem, quia voluptas sit, aspernatur aut odit aut fugit, sed quia
consequuntur magni dolores eos, qui ratione voluptatem sequi nesciunt,
neque porro quisquam est, \textbf{qui do lorem ipsum, quia dolor sit amet}
consectetur adipisci[ng] velit, sed quia non numquam [do] eius modi tempora
inci[di]dunt, ut labore et dolore magnam aliquam quaerat voluptatem. Ut enim
ad minima veniam, quis nostrum[d] exercitationem ullam corporis suscipit
laboriosam, nisi ut aliquid ex ea commodi consequatur? [D]Quis autem vel eum
iure reprehenderit, qui in ea voluptate velit esse, quam nihil molestiae
consequatur, vel illum, qui dolorem eum fugiat, quo voluptas nulla pariatur?
At vero eos et accusamus et iusto odio dignissimos ducimus, qui blanditiis
praesentium voluptatum deleniti atque corrupti, quos dolores et quas molestias
excepturi sint, obcaecati cupiditate non provident, similique sunt in culpa, qui
officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum
quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum
soluta nobis est eligendi optio, cumque nihil impedit, quo minus id, quod
maxime placeat, facere possimus, omnis voluptas assumenda est, omnis dolor
repellendus. Temporibus autem quibusdam et aut officiis debitis aut rerum
necessitatibus saepe eveniet, ut et voluptates repudiandae sint et molestiae
non recusandae. Itaque earum rerum hic tenetur a sapiente delectus, ut aut
reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus
asperiores repellat.
\end{latinpage}
\end{document}
pdftk. – Marijn Jan 16 '21 at 17:31changepagepackage and its\checkoddpagemacro. – Peter Wilson Jan 16 '21 at 17:50