  We give a new construction of a Hopf algebra defined first by Reading whose
bases are indexed by objects belonging to the Baxter combinatorial family
(i.e., Baxter permutations, pairs of twin binary trees, etc.). Our construction
relies on the definition of the Baxter monoid, analog of the plactic monoid and
the sylvester monoid, and on a Robinson-Schensted-like correspondence and
insertion algorithm. Indeed, the Baxter monoid leads to the definition of a
lattice structure over pairs of twin binary trees and the definition of a Hopf
algebra. The algebraic properties of this Hopf algebra are studied and among
other, multiplicative bases are provided, and freeness and self-duality proved.
