Let G be a real reductive Lie group, and Hℂ the complexification of its maximal compact subgroup H⊂G. We consider classes of semistable G-Higgs bundles over a Riemann surface X of genus g≥2 whose underlying Hℂ-principal bundle is unstable. This allows us to find obstructions to a deformation retract from the moduli space of G-Higgs bundles over X to the moduli space of Hℂ-bundles over X, in contrast with the situation when g=1, and to show reducibility of the nilpotent cone of the moduli space of G-Higgs bundles, for G complex.