  Spinon-magnon mixing was recently reported in botallackite Cu$_2$(OH)$_3$Br
with a uniaxially compressed triangular lattice of Cu$^{2+}$ quantum spins
[Zhang et al., Phys. Rev. Lett. 125, 037204 (2020)]. Its nitrate counterpart
rouaite, Cu$_2$(OH)$_3$NO$_3$, 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
ferro- 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$^\circ$ 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.
