Parametrizing beyond Standard Model physics by the SU(3)×SU(2)L×U(1)Y dimension-six effective lagrangian, we study the impact of anomalous Higgs couplings in angular asymmetries of the crossing symmetric processes H→Zℓ+ℓ− and e+e−→HZ. In the light of present bounds on d = 6 couplings, we show that some asymmetries can reveal BSM effects that would otherwise be hidden in other observables. The d=6 HZγ couplings as well as (to a lesser extent) HZℓℓ contact interactions can generate asymmetries at the several percent level, albeit having less significant effects on the di-lepton invariant mass spectrum of the decay H→Zℓ+ℓ−. The higher di-lepton invariant mass probed in e+e−→HZ can lead to complementary anomalous coupling searches at e+e− colliders.