The influence of radial and poloidal ExB drifts in the Scrape-Off Layer of a divertor tokamak are considered. For B→ in the "forward" direction (the ∇→B ion drift toward a lower X-point, single null divertor configuration), the poloidal drift tends to increase the plasma pressure and the recycling rates at the outer target, lowering the temperature there - as compared with the inner target. When the radial drift dominates, these asymmetries are reversed. The radial drift is estimated to be more important than the poloidal drift for high ne-operation (the "high recycling" regime). Poloidal drifts can result in substantially reduced plasma contact at one target ("detachment"); simultaneous detachment at both targets due to poloidal drifts appears to be less likely. Radial drifts could result in detachment at one or both targets, but the type of detachment involved does not appear to be consistent with that experimentally observed. The experimental observations made in the Tokamak de Varennes, with divertor biasing arranged to increase/decrease the radial electric field in the SOL, relative to its natural level, are broadly consistent with the poloidal drift model.