  The multiplihedra {M_n} form a family of polytopes originating in the study
of higher categories and homotopy theory. While the multiplihedra may be
unfamiliar to the algebraic combinatorics community, it is nestled between two
families of polytopes that certainly are not: the permutahedra {S_n} and
associahedra {Y_n}. The maps between these families reveal several new Hopf
structures on tree-like objects nestled between the Malvenuto-Reutenauer (MR)
Hopf algebra of permutations and the Loday-Ronco (LR) Hopf algebra of planar
binary trees. We begin their study here, constructing a module over MR and a
Hopf module over LR from the multiplihedra. Rich structural information about
this module is uncovered via a change of basis--using M\"obius inversion in
posets built on the 1-skeleta of the {M_n}. Our analysis uses the notion of an
interval retract, which should have independent interest in poset
combinatorics. It also reveals new families of polytopes, and even a new
factorization of a known projection from the associahedra to hypercubes.
