New data is presented on K&pgr;= 0+bands in156,158Gd and178Hf. Each of these nuclei exhibit a different aspect of the nature of K&pgr;= 0+bands in deformed nuclei. In156Gd, there is evidence of a K&pgr;= 0+band with significant two‐phonon &ggr;&ggr; component whereas in178Hf results are shown for a K&pgr;= 0+band whose members exhibit two‐phonon &bgr;&bgr; character. New (p,t) reaction results point to the existence of thirteen excited 0+states in158Gd below an excitation energy of 3.1 MeV. All the results combine to provide a challenge to our present understanding of the nature of these excitations. © 2002 American Institute of Physics