We report in-plane resistivity (ρ) and transverse magnetoresistance (MR) measurements for underdoped HgBa2CuO4+δ (Hg1201). Contrary to the long-standing view that Kohler's rule is strongly violated in underdoped cuprates, we find that it is in fact satisfied in the pseudogap phase of Hg1201. The transverse MR shows a quadratic field dependence, δρ/ρ0=aH2, with a(T)T-4. In combination with the observed ρT2 dependence, this is consistent with a single Fermi-liquid quasiparticle scattering rate. We show that this behavior is typically masked in cuprates with lower structural symmetry or strong disorder effects.