When two persons anger with each other,there is a long distance between their hearts.So they quarrel loudly to fill it.It's the reason for lovers talking in low voice.
I can anger you, and you can anger me. But two people can not "anger with each other."
The 'get angry with each other.'
PS So the pattern is asymmetrical:
I anger you [='You get angry with me because of something I've done.']
You anger me [='I get angry with you because of something you've done.']
no equivalent We get angry with each other.