畳む and 裏返す: Folding objects and turning them over
Compare folding a shirt or umbrella with turning over meat or paper. A map seen after the paper is flipped was already printed on the other side.
畳む as folding
畳む（たたむ） means to fold. In 「乾いたシャツをきれいに畳んで棚に入れた。」, the speaker neatly folded a dry shirt and put it on a shelf. Folding is followed by putting it away, and the shirt is already dry before it is folded.
In 「雨がやんだので、傘を畳む。」, the speaker folds the umbrella because the rain has stopped. The end of the rain gives the reason for folding it.
裏返す as turning over
裏返す（うらがえす） means to turn over or flip. In 「片面がこんがり焼けた肉を裏返した。」, the speaker turned over meat after one side had browned. The action changes which side faces up.
In 「紙を裏返すと地図が印刷されていた。」, turning over the paper revealed a printed map. The flip exposed the other side; the map was already printed there.
A folded shape and the opposite face
畳む describes folding the shirt or umbrella, while 裏返す describes changing the exposed side of the meat or paper. Look for a folded shape in the first action and a change of face in the second.
In the paper example, と leads to the state found after the flip. 地図が印刷されていた describes an existing printed map, so the act of flipping the paper did not print the map.
In 紙を裏返すと地図が印刷されていた, did turning over the paper cause a new map to be printed?
Show the answer and why
No. Turning over the paper revealed a map that was already printed on the reverse side. 裏返す describes the flip, not the creation of the printed map.