In recent measurements of the scissors mode in radiative decay experiments, transition strengths were observed that were double that expected from theory and systematics well established from measurements on the radiative excitation channel, that is, using nuclear resonance fluorescence (NRF). Additional strength as measured with NRF can only be present as heretofore unobserved branching or fragmentation of the scissors mode. Such possibilities were investigated in a transmission NRF measurement on the deformed, odd-mass 181 Ta, using a quasi-monoenergetic γ-ray beam at two beam energies. This measurement further influences applications using transmission NRF to assay or detect odd-mass fissile isotopes. A large branching, ≈75%, of small resonances to excited states was discovered. In contrast, previous studies using NRF of the scissors-mode strength in odd-mass nuclei assumed no branching existed.The presently observed branching, combined with the observed highly-fragmented elastic strength, could reconcile the scissors-mode strength observed in NRF measurements with the expectations for enhanced scissors-mode strength from radiative decay experiments. Recently, however, in a measurement probing the radiative-decay channel, where decay γ-rays were measured following excitation of the nucleus by light ion-scattering, double the expected scissors-mode strength was observed in both even and odd-mass actinide nuclei compared to expectations established for the radiative excitation channel, that is, using NRF [6]. The same phenomenon of increased strength was observed a decade prior in lanthanide nuclei using similar measurements of radiative decay following neutron capture [7,8] and ( 3 He, α) scattering [8], which observation has been reinforced in a contemporary measurement. This discrepancy in measured strength between the radiative excitation and radiative decay channels for the scissors mode remains its greatest mystery.Additional scissors-mode strength can only be present in the radiative excitation channel if it were to occur as heretofore unobserved fragmentation or branching, forbearing the alternative that the additional strength is confined solely to the radiative decay channel. This alternative could happen if states excited by ion-scattering at energies up to 10 MeV de-excite emitting 2-3 MeV γ-rays more strongly than would be expected from the ground state scissors-mode strength (see e.g. Krtička et al. [7]). Nevertheless, the possibility of strong branching of the scissors mode to excited states was recently suggested from theoretical calculations [10]. To date, such branching of the scissors mode has not been investigated, and it was assumed by necessity to be negligible in previous attempts with NRF to investigate the total strength and fragmentation of the scissors mode in odd-mass nuclei [5]. The observation of additional scissorsmode strength using NRF, as branching or fragmentation, would not only challenge theoretical models of the scissors mode, but also impact proposed applications t...