  We generalize Bruned et.al.'s notion of translation in geometric and branched
rough paths to a notion of translation in rough paths over any combinatorial
Hopf algebra. We show that this notion of translation is equivalent to two
bialgebras being in cointeraction, subject to certain additional conditions. We
argue that reformulating translations in terms of substitutions, provides
simpler conditions for the cointeraction formulation. For the special case
where the translation can be obtained from a product, we show how to obtain a
description of the dual coaction. As a concrete example, we describe
translations in planarly branched rough paths.
