  We consider bifurcation of critical points from a trivial branch for families
of functionals that are invariant under the orthogonal action of a compact Lie
group. Based on a recent construction of an equivariant spectral flow by the
authors, we obtain a bifurcation theorem that generalises well-established
results of Smoller and Wasserman as well as Fitzpatrick, Pejsachowicz and
Recht. Finally, we discuss some elementary examples of strongly indefinite
systems of PDEs and Hamiltonian systems where the mentioned classical
approaches fail but an invariance under an orthogonal action of $\mathbb{Z}_2$
makes our methods applicable and yields the existence of bifurcation.
