The article gets it right, but "man muss immer umkehren," is better translated as "man must always turn upside down", "inside out," "turn back" or "reverse" depending on the context.
In Afrikaans, "omkeer" is derived from the Germanic umkehren and would be used as changing direction (in a military sense) or upside down as in 'leave no stone unturned.'
Strangely, nowadays I would refer to inverting your trousers as "binneste-buite" (inside-out) or "uitkeer" in Afrikaans: roughly 'about face'.
To turn a piece of clothing inside out is "auf Links ziehen", which is hard to translate. I find "inside out" to be quite the intuitive metaphor.
The idea is that clothes have a "right" side (rechts) and a "left" side (links), and you pull it (ziehen) so that the left side is visible, i.e. on the outside.
Someone wrote that the terms left and right come from knitting where the right side is the flat side, and thus worn on the outside. Not sure whether that holds water.
According to the wikipedia page about Jacobi, in the mathematical context the best translation is invert.
Also in Italian (my language)it is used the verb 'invertire' in this context.
In Afrikaans, "omkeer" is derived from the Germanic umkehren and would be used as changing direction (in a military sense) or upside down as in 'leave no stone unturned.'
Strangely, nowadays I would refer to inverting your trousers as "binneste-buite" (inside-out) or "uitkeer" in Afrikaans: roughly 'about face'.