  We construct recently introduced palatial NC twistors by considering the pair
of conjugated (Born-dual) twist-deformed $D=4$ quantum inhomegeneous conformal
Hopf algebras $\mathcal{U}_{\theta }(su(2,2)\ltimes T^{4}$) and
$\mathcal{U}_{\bar{\theta}}(su(2,2)\ltimes\bar{T}^{4}$), where $T^{4}$ describe
complex twistor coordinatesand $\bar{T}^{4}$ the conjugated dual twistor
momenta. The palatial twistors are suitably chosen as the quantum-covariant
modules (NC representations) of the introduced
  Born-dual Hopf algebras. Subsequently we introduce the quantum deformations
of $D=4$ Heisenberg-conformal algebra (HCA) $su(2,2)\ltimes H^{4,4}_\hslash$
($H^{4,4}_\hslash=\bar{T}^4 \ltimes_\hslash T_4$ is the Heisenberg algebra of
twistorial oscillators) providing in twistorial framework the basic covariant
quantum elementary system.
  The class of algebras describing deformation of HCA with dimensionfull
deformation parameter, linked with Planck length $\lambda_p$ will be called the
twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in
space-time framework.
  We shall describe the examples of TDSR algebra linked with Palatial twistors
which are introduced by the Drinfeld twist and by the quantization map in
$H_\hslash^{4,4}$. We introduce as well generalized quantum twistorial phase
space by considering the Heisenberg double of Hopf algebra
$\mathcal{U}_\theta(su(2,2)\ltimes T^4).$
