New probes of neutrino mixing are needed to advance precision studies. One promising direction is via the detection of low-energy atmospheric neutrinos (below a few hundred MeV), to which a variety of near-term experiments will have much-improved sensitivity. Here we focus on probing these neutrinos through distinctive nuclear signatures of exclusive neutrino-carbon interactions — those that lead to detectable nuclear-decay signals with low backgrounds — in both neutral-current and charged-current channels. Here the neutral-current signature is a line at 15.11 MeV and the charged-current signatures are two- or three-fold coincidences with delayed decays. We calculate the prospects for identifying such events in the Jiangmen Underground Neutrino Observatory (JUNO), a large-scale liquid-scintillator detector. A five-year exposure would yield about 16 neutral-current events (all flavors) and about 16 charged-current events (mostly from , with some from ), and thus roughly 25\% uncertainties on each of their rates. Our results show the potential of JUNO to make the first measurement of sub-100 MeV atmospheric neutrinos. They also a step towards multi-detector studies of low-energy atmospheric neutrinos, including with the goal of identifying additional distinctive nuclear signatures for carbon and other targets.