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To Heisuke Hironaka, on the occasion of his 80th birthday Abstract. A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially of finite type over a noetherian scheme S. The theory takes values in the category of symmetric graded modules over the graded-commutative ring ⊕iHi(S,OS). In degree i, the cohomology and homology H0(S,OS)-modules thereby associated to such an x: X → S, with Hochschild complex Hx, are ExtiOX(Hx,Hx) and Ext−iOX(Hx, x!OS) (i ∈ Z). This lays the foundation for a sequel that will treat orientations in bivariant Hochschild theory through canonical relative fundamental class maps, unifying and generalizing previousl