We study the full angular distribution of semileptonic rare charm baryon decays in which the secondary baryon undergoes weak decay with sizable polarization parameter, Ξc+→Σ+(→p π0) ℓ+ℓ- , Ξc0→Λ0(→p π-) ℓ+ℓ- and Ωc0→Ξ0(→Λ0π0) ℓ+ℓ- . Such self-analyzing decay chains allow for seven additional observables compared to three-body decays such as Λc→p ℓ+ℓ- , with different sensitivities to the |Δ c |=|Δ u |=1 weak couplings. Opportunities to test the standard model in c →u transitions with standard model null tests and other angular observables are worked out. We show that a joint model-independent analysis of the leptonic AFBℓ, hadronic AFBH, and combined forward-backward asymmetry AFBℓ H together with the fraction of longitudinally produced leptons, FL, is able to pin down the dipole couplings C7,C7' and the semileptonic (axial-) vector ones C10,C9',C10' . AFBH is also accessible with dineutrino c →u ν ν ¯ modes and probes right-handed currents.