Complementary Hamiltonian cycle in hypercube

Question

I want to prove that every hypercube $Q_n$ where $n$ is even has a cycle where the vertices with the greatest possible distance ($n$) in the hypercube also have the greatest possible distance on the cycle ($2^{n-1}$).

I found such an example for $Q_4$ ($Q_2$ is obvious) but I don't know how to generalize the idea to all even $n$.