It seems to have a strong sense around "returning to a previous place / state", not just "turn around". Which is where the mathematical use of "invert" comes from, because an inverted function swaps the domain and range. But also words like "repent" being a possible translation; to repent of your actions and thus (ideally) return to a previous state of innocence.
https://dict.leo.org/german-english/umkehren
It seems to have a strong sense around "returning to a previous place / state", not just "turn around". Which is where the mathematical use of "invert" comes from, because an inverted function swaps the domain and range. But also words like "repent" being a possible translation; to repent of your actions and thus (ideally) return to a previous state of innocence.