We consider the generalized (10+10)-dimensional D=4 quantum phase spaces containing translational and Lorentz spin sectors associated with the dual pair of twist-quantized Poincare Hopf algebra $mathbb{H}$ and quantum Poincare Hopf group $widehat{mathbb{G}}$. Two Hopf algebroid structures of generalized phase 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 space containing twisted noncommutative Minkowski space $hat{x}_{mu }$. We obtain the first explicit example of Hopf algebroid structure of relativistic quantum phase space which contains quantum-deformed Lorentz spin sector.
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