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The electronic character of photoexcited molecules can abruptly change at avoided crossings and conical intersections. Here, we report direct mapping of the coupled interplay between electrons and nuclei in a prototype molecule, iodine monobromide (IBr), by using attosecond transient absorption spectroscopy. A few-femtosecond visible pulse resonantly excites the (B3P0(+)), Y(0(+)), and Z(0(+)) states of IBr, and the photodissociation dynamics are tracked with an attosecond extreme-ultraviolet pulse that simultaneously probes the I-4d and Br-3d core-level absorption edges. Direct comparison with quantum mechanical simulations unambiguously identifies the absorption features associated with adiabatic and diabatic channels at the B/Y avoided crossing and concurrent two-photon dissociation processes that involve the Y/Z avoided crossing. The results show clear evidence for rapid switching of valence-electronic character at the avoided crossing.