We consider the twisting of the Hopf structure for the classical enveloping algebra U(g), where g is an inhomogenous rotation algebra, with explicit formulae given for the D=4 Poincare algebra (g=P4). The comultiplications of twisted UF(P4) are obtained by conjugating the primitive classical coproducts by F in U(C)(X)U(C), where c denotes any Abelian subalgebra of P4, and the universal R-matrices for UF(P4) are triangular. As an example we show that the quantum deformation of the Poincare algebra recently proposed by Chaichian and Demiczev is a twisted classical Poincare algebra. The interpretation of the twisted Poincare algebra as describing relativistic symmetries with clustered two-particle states is proposed.