  We consider two new classes of twisted D=4 quantum Poincar\'{e} symmetries
described as the dual pairs of noncocommutative Hopf algebras. Firstly we
investigate a two-parameter class of twisted Poincar\'{e} algebras which
provide the examples of Lie-algebraic noncommutativity of the translations. The
corresponding associative star-products and new deformed Lie-algebraic
Minkowski spaces are introduced. We discuss further the twist deformations of
Poincar\'{e} symmetries generated by the twist with its carrier in Lorentz
algebra. We describe corresponding deformed Poincar\'{e} group which provides
the quadratic deformations of translation sector and define the quadratically
deformed Minkowski space-time algebra.
