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We consider the generalized (10+10)-dimensional D=4 quantum phase spacescontaining translational and Lorentz spin sectors associated with the dual pairof twist-quantized Poincare Hopf algebra $\mathbb{H}$ and quantum Poincare Hopfgroup $\widehat{\mathbb{G}}$. Two Hopf algebroid structures of generalizedphase spaces with spin sector will be investigated: first one $%\mathcal{H}^{(10,10)}$ describing dynamics on quantum group algebra $%\widehat{\mathbb{G}}$ provided by the Heisenberg double algebra $\mathcal{HD=%}\mathbb{H}\rtimes \widehat{\mathbb{G}}$, and second, denoted by $\mathcal{%\tilde{H}}^{(10,10)}$, describing twisted Hopf algebroid with base spacecontaining twisted noncommutative Minkowski space $\hat{x}_{\mu }$. We obtainthe first explicit example of Hopf algebroid structure of relativistic quantumphase space which contains quantum-deformed Lorentz spin sector.

Quantum Twist-Deformed D=4 Phase Spaces with Spin Sector and Hopf Algebroid Structures