Picture two arrows pointing at each other instead of one arrow pointing one way. Whatever the sentence says next, mutually says it travels in both directions: each side does it to the other, or both sides get it. That is the whole job of the word, and everything else it appears in is just that pair of arrows put to different use.
mutually 可以想成兩支箭頭對著彼此，而不是一支箭頭指向一邊。後面接什麼，它就把那件事變成雙向的：你對我、我也對你，或者兩邊一起被影響到。這個字做的事情就只有這一件。它出現在哪個說法裡，都只是那對箭頭換了個用途。
From the word’s history
The packet's Wiktionary etymology gives mutually as mutual + -ly: the adjective for something two sides share, turned into a word that describes how an action or state works.
來源的字源說明是 mutual 加 -ly。mutual 指「兩邊都有份的」，-ly 把它變成描述動作或狀態的副詞，所以 mutually 就是「以兩邊都有份的方式」——正好就是那對互指的箭頭。
packet Wiktionary etymology_text for mutually (kaikki:3f6f69a190036ae4:1057346198): From mutual + -ly.