Frustrated magnets can elude the paradigm of conventional symmetry breaking and instead exhibit signaturesof emergent symmetries at low temperatures. Such symmetries arise from “accidental” degeneracies within theground state manifold and have been explored in a number of disparate models, in both two and three dimen-sions. Here we report the systematic construction of a family of classical spin models that, for a wide varietyof lattice geometries with triangular motifs in one, two and three spatial dimensions, such as the kagome or hy-perkagome lattices, exhibit an emergent, continuous U(1) symmetry. This is particularly surprising because theunderlying Hamiltonian actually has very little symmetry — a bond-directional, off-diagonal exchange modelinspired by the microscopics of spin-orbit entangled materials (the Γ′-model). The construction thus allowsfor a systematic study of the interplay between the emergent continuous U(1) symmetry and the underlyingdiscrete Hamiltonian symmetries in different lattices across different spatial dimensions. We discuss the impactof thermal and quantum fluctuations in lifting the accidental ground state degeneracy via the thermal and quan-tum order-by-disorder mechanisms, and how spatial dimensionality and lattice symmetries play a crucial rolein shaping the physics of the model. Complementary Monte Carlo simulations, for representative one-, two-,and three-dimensional lattice geometries, provide a complete account of the thermodynamics and confirm ouranalytical expectations.