The Hamiltonian Hopf bifurcation has an integrable normal form that describes
the passage of the eigenvalues of an equilibrium through the 1 : −1 resonance. At
the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium turn into a
complex quadruplet of eigenvalues and the equilibrium becomes a linearly unstable
focus-focus point. We explicitly calculate the frequency map of the integrable normal
form, in particular we obtain the rotation number as a function on the image of the
energy-momentum map in the case where the fibres are compact. We prove that
the isoenergetic non-degeneracy condition of the KAM theorem is violated on a curve
passing through the focus-focus point in the image of the energy-momentum map. This
is equivalent to the vanishing of twist in a Poincar´e map for each energy near that of
the focus-focus point. In addition we show that in a family of periodic orbits (the non￾linear normal modes) the twist also vanishes. These results imply the existence of all
the unusual dynamical phenomena associated to non-twist maps near the Hamiltonian
Hopf bifurcation.