We characterise the intermediate logics which admit a cut-free hypersequent calculus of the form \, where \ is the hypersequent counterpart of the sequent calculus \ for propositional intuitionistic logic, and \ is a set of so-called structural hypersequent rules, i.e., rules not involving any logical connectives. The characterisation of this class of intermediate logics is presented both in terms of the algebraic and the relational semantics for intermediate logics. We discuss various—positive as well as negative—consequences of this characterisation.