Drinking water, sanitation and hygiene behaviour, referred to as the wash variables by the United Nations Children's Emergency Fund, are acknowledged as the three main determinants of diarrhoeal diseases. But the impact of their complementarities on disease incidence remains understudied. This study uses state and household level data to examine the determinants of child diarrhoeal incidence. It introduces indicators of wash quality and combined presence, both atthe household and state levels. It combines them in a novel analysis to understand their roles. In the Indian states, with the worst wash infrastructure, these variables are strategic substitutes, but as wash infrastructure improves, they become strategic complements. Thus, resource allocation to lower diarrhoea incidence must take into account the complementary rather than individual presence of these focal variables. Further, the quality of wash also matters. The Swachh Bharat Abhiyan, targeting universal sanitation coverage, is unlikely to be effective unless it breaks the Gordian knot of complementarities and wash quality holding up the burden of childhood diarrhoea.