there seems to be some sort of translation anomaly happening ...

I couldn't find any reference to chasseurs a cheval being employed as mounted infantry ... finally I tried looking for the equivalent in other languages, and found this on German Wikipedia -

"... Jager zu Pferde (franzosisch: Chasseurs a Cheval, englisch: Mounted Riflemen bzw. Mounted Rifles) ..."

As near as I can figure out, there is a school of thought, not confined to France or to the publisher's blurb for the new book, that since chasseurs a pied = "riflemen" (approximately, sort of, sometimes, etc.) chasseurs a cheval must be translatable as "mounted riflemen" or in the present case "mounted infantry".