
beidseitig
on both sides
Located or taking place on both sides of something.
Bitte drucken Sie das Dokument beidseitig, um Papier zu sparen.
Please print the document on both sides to save paper.
on both sides / mutual
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on both sides
Located or taking place on both sides of something.
Bitte drucken Sie das Dokument beidseitig, um Papier zu sparen.
Please print the document on both sides to save paper.

mutual
Concerning two parties or sides; mutual or reciprocal. Often used in connection with contracts or agreements.
Die Kündigung des Vertrags erfolgte in beidseitigem Einvernehmen.
The contract was terminated by mutual agreement.
The word is made up of 'beid-' (from 'beide', both) and '-seitig' (from 'Seite', side). So it literally means 'on both sides' or 'concerning both sides'.
Die Verhandlungen erforderten beidseitige Kompromisse.
The negotiations required mutual compromises.
'Beidseitig' (on both sides/mutual) can stand not only as an adjective before a noun, but also as an adverb describing a verb. In this function, it often comes at the end of the sentence.
Das Formular muss beidseitig ausgefüllt werden.
The form must be filled out on both sides.
You can recognize the meaning from the context. If it is about physical objects (paper, road), it means 'on both sides'. If it is about abstract things (contracts, feelings), it means 'mutual'.
Die Straße ist wegen Bauarbeiten nur einseitig befahrbar, nicht beidseitig.
Because of construction work, the road is only passable on one side, not on both sides.
In the meaning of 'mutual', 'beidseitig' is often used in formal or legal contexts. Typical combinations are 'beidseitiges Einvernehmen' (mutual agreement), 'beidseitiges Interesse' (mutual interest), or 'beidseitiger Respekt' (mutual respect).
Der Arbeitsvertrag wurde in beidseitigem Einvernehmen aufgelöst.
The employment contract was terminated by mutual agreement.
mutual
Gegenseitig and beidseitig can refer to two parties. Gegenseitig emphasizes that an action or obligation is directed toward each other; beidseitig merely highlights that both sides are affected. Beidseitig is suitable for obligations or agreements, while gegenseitig is used for an action done for each other.

Der Vertrag enthält beidseitige Pflichten für die beiden Vertragspartner.
The contract contains obligations for both parties.
Die beiden Vertragspartner unterstützen sich gegenseitig bei der Umsetzung des Projekts.
The two parties support each other in carrying out the project.