The nature of the pseudogap phase of the copper oxides ('cuprates') remains a puzzle. Although there are indications that this phase breaks various symmetries, there is no consensus on its fundamental nature(1). Fermi-surface, transport and thermodynamic signatures of the pseudogap phase are reminiscent of a transition into a phase with antiferromagnetic order, but evidence for an associated long-range magnetic order is still lacking(2). Here we report measurements of the thermal Hall conductivity (in the x-y plane, kappa(xy)) in the normal state of four different cuprates-La1.6-xNd0.4SrxCuO4, La1.8-xEu0.2SrxCuO4, La2-xSrxCuO4 and Bi2Sr2-xLaxCuO6+delta. We show that a large negative kappa(xy) signal is a property of the pseudogap phase, appearing at its critical hole doping, p*. It is also a property of the Mott insulator at p approximate to 0, where kappa(xy) has the largest reported magnitude of any insulator so far(3). Because this negative kappa(xy) signal grows as the system becomes increasingly insulating electrically, it cannot be attributed to conventional mobile charge carriers. Nor is it due to magnons, because it exists in the absence of magnetic order. Our observation is reminiscent of the thermal Hall conductivity of insulators with spin-liquid states(4-6), pointing to neutral excitations with spin chirality(7) in the pseudogap phase of cuprates.