It seems to me that a better English equivalent to umkehren would be turn around. That also fits the text of the article better. That is to say look at the problem from the other side, form another angle. Invert is simultaneously too specific (leading to formulaic methods) and too ambiguous (leading to pointless discussions of whether it means considering putting the steering wheel at the back or on the other side when in fact it means do both).
> He is said to have told his students that when looking for a research topic, one should 'Invert, always invert' ('man muss immer umkehren'), reflecting his belief that inverting known results can open up new fields for research, for example inverting elliptical integrals and focusing on the nature of elliptic and theta functions.[8]
> The great mathematician Jacobi is said to have
inculcated upon his students the dictum: Man muss immer
umkehren. One must always seek a converse, turn a thought
the other end to.
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.
See https://en.wiktionary.org/wiki/umkehren
Edit: fixed typo.