
gegeneinander
against each other
Expresses that two or more parties act as opponents in a competition, fight, or dispute.
Im Finale spielten die beiden besten Mannschaften gegeneinander.
In the final, the two best teams played against each other.
against each other
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against each other
Expresses that two or more parties act as opponents in a competition, fight, or dispute.
Im Finale spielten die beiden besten Mannschaften gegeneinander.
In the final, the two best teams played against each other.

into each other
Describes a movement in which two or more objects collide with each other or hit each other from opposite directions.
Die beiden Autos sind frontal gegeneinander gefahren.
The two cars drove head-on into each other.
The word is made up of 'gegen' (against) and 'einander' (each other). So it describes an action that two or more parties direct toward each other.
Die Kinder warfen den Ball gegeneinander.
The children threw the ball at each other.
Since 'gegeneinander' (against each other) expresses a reciprocal relationship, the subject of the sentence must always be plural or refer to a group.
Die beiden Teams treten morgen gegeneinander an.
The two teams will compete against each other tomorrow.
In the context of competition or dispute, 'gegeneinander' (against each other) is often used with verbs like 'spielen' (to play), 'kämpfen' (to fight), 'antreten' (to compete), or 'argumentieren' (to argue).
Im Debattierclub argumentieren die Mitglieder oft gegeneinander.
In the debate club, the members often argue against each other.
When referring to a collision, 'gegeneinander' (into each other) is used with verbs of movement. Typical examples are 'fahren' (to drive), 'laufen' (to run), 'prallen' (to crash), or 'stoßen' (to bump).
Auf der glatten Straße sind zwei Autos gegeneinander gerutscht.
On the slippery road, two cars slid into each other.
on each other / one after another
Aufeinander describes mutual alignment in movement, for example when two vehicles are moving toward each other. Gegeneinander emphasizes contact or a collision in opposite directions. Before the collision, they move aufeinander zu (toward each other); at the moment of impact, they collide gegeneinander.

Kurz darauf prallen die beiden Einkaufswagen gegeneinander.
Shortly afterward, the two shopping carts collide with each other.
Die beiden Einkaufswagen rollen ungebremst aufeinander zu.
The two shopping carts roll toward each other without slowing down.