Each-other

pronoun
1.
each the other; one another (used as a compound reciprocal pronoun):
to strike at each other; to hold each other’s hands; to love each other.
pronoun
1.
used when the action, attribution, etc, is reciprocal: furious with each other

reciprocal pronoun, originally in late Old English a phrase, with each as the subject and other inflected (as it were “each to other,” “each from other,” etc.).
Also, one another. Each one the other, one the other, as in The boys like each other, or The birds were fighting one another over the crumbs. Both of these phrases indicate a reciprocal relationship or action between the subjects preceding (the boys, the birds). Formerly, many authorities held that each other should be confined to a relationship between two subjects only and one another used when there are more than two. Today most do not subscribe to this distinction, which was never strictly observed anyway. [ Late 1300s ]
Also see: at each other’s throats