  The $(4+4)$-dimensional $\kappa$-deformed quantum phase space as well as its
$(10+10)$-dimensional covariant extension by the Lorentz sector can be
described as Heisenberg doubles: the $(10+10)$-dimensional quantum phase space
is the double of $D=4$ $\kappa$-deformed Poincar\'e Hopf algebra $\mathbb{H}$
and the standard $(4+4)$-dimensional space is its subalgebra generated by
$\kappa$-Minkowski coordinates $\hat{x}_\mu$ and corresponding commuting
momenta $\hat{p}_\mu$. Every Heisenberg double appears as the total algebra of
a Hopf algebroid over a base algebra which is in our case the coordinate
sector. We exhibit the details of this structure, namely the corresponding
right bialgebroid and the antipode map. We rely on algebraic methods of
calculation in Majid-Ruegg bicrossproduct basis. The target map is derived from
a formula by J-H. Lu. The coproduct takes values in the bimodule tensor product
over a base, what is expressed as the presence of coproduct gauge freedom.
