We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane H. The angles of lattice points arising from the orbit of the modular group PSL2(Z), and lying on hyperbolic circles, are shown to be equidistributed for generic radii. However, the angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of Z2-lattice points (with certain parity conditions) lying on circles in R2, along a thin subsequence of radii. A notable difference is that measures in the hyperbolic setting can break symmetry; on very thin subsequences they are not invariant under rotation by π/2, unlike in the Euclidean setting where all measures have this invariance property.