We introduce the theory IHR of interacting Hopf algebras, parametrised over a principal ideal domain 134b" title="Click to view the MathML source">R. The axioms of IHR are derived using Lack's approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoid–comonoid pairs. This construction is instrumental in showing that IHR is isomorphic to the PROP of linear relations (i.e. subspaces) over the field of fractions of 134b" title="Click to view the MathML source">R.