Given a boolean $0/1$ cube in $n$ dimensions with $2^{n-1}$ red and $2^{n-1}$ blue points can we complement the cube (blue becomes red and vice versa) in $\operatorname{poly}(n)$ transformations?

Here by transformation I mean the following.

Cutting the cube by $h=\operatorname{poly}(n)$ hyperplane inequalities each describable by $poly(n)$ bit coefficients into $r=\operatorname{poly}(n)$ pieces $P_1,\dots,P_r$.

Permuting coordinates of each piece (for example if $101$ was blue and $011$ was red then permuting first and second coordinate $011$ is blue and $101$ is blue).

Negating coordinates of each piece (for example if $101$ was blue and $111$ was red then negating first coordinate $111$ is blue and $101$ is blue).

Rotating, translating each piece.

Finally repasting all pieces so it becomes cube again.