Three new highly-deformed (HD) bands are identified in Nd136 and the highly deformed band of Nd137 is extended at higher spin by four transitions, revealing a band crossing associated with the occupation of the second νi13/2 intruder orbital. Extended cranked Nilsson-Strutinsky calculations are performed for all HD bands observed in Nd134, Nd136, and Nd137, achieving for the first time a consistent interpretation of all HD bands in the Nd nuclei. The new interpretation has significant consequences, like the change of parity of the yrast HD bands of Nd134 and Nd136, and the involvement of two negative-parity neutron intruder orbitals in the configurations of most HD bands. The present experimental results and their theoretical interpretation represent an important step forward in the understanding of the second-minimum excitations in the Nd nuclei.