We propose two models where a U(1) Peccei-Quinn global symmetry arises accidentally and is respected up to high-dimensional operators, so that the axion solution to the strong CP problem is successful even in the presence of Planck-suppressed operators. One model is SU() gauge interactions with fermions in the fundamental and a scalar in the symmetric. The axion arises from spontaneous symmetry breaking to SO(), that confines at a lower energy scale. Axion quality in the model needs ≳ 10. SO bound states and possibly monopoles provide extra Dark Matter candidates beyond the axion. In the second model the scalar is in the anti-symmetric: SU() broken to Sp() needs even ≳ 20. The cosmological DM abundance, consisting of axions and/or super-heavy relics, can be reproduced if the PQ symmetry is broken before inflation (Boltzmann-suppressed production of super-heavy relics) or after (super-heavy relics in thermal equilibrium get partially diluted by dark glue-ball decays).