Spinon-magnon mixing was recently reported in botallackite Cu_2⁢(OH)₃⁢Br with a uniaxially compressed triangular lattice of Cu²⁺ quantum spins [H. Zhang et al., Phys. Rev. Lett. 125, 037204 (2020)]. Its nitrate counterpart rouaite, Cu₂⁢(OH)₃⁢NO₃, has a highly analogous structure and might be expected to exhibit similar physics. To lay a foundation for research on this material, we clarify rouaite's magnetic phase diagram and identify both low-field phases. The low-temperature magnetic state consists of alternating ferromagnetic and antiferromagnetic chains, as in botallackite, but with additional canting, leading to net moments on all chains which rotate from one chain to another to form a 90∘ cycloidal pattern. The higher-temperature phase is a helical modulation of this order, wherein the spins rotate from one Cu plane to the next. This extends to zero temperature for fields perpendicular to the chains, leading to a set of low-temperature field-induced phase transitions. Rouaite may offer another platform for spinon-magnon mixing, while our results suggest a delicate balance of interactions and high tunability of the magnetism.