We use the shear construction to construct and classify a wide range of two-step solvable Lie groups admitting a left-invariant SKT structure. We reduce this to a specification of SKT shear data on Abelian Lie algebras, and which then is studied more deeply in different cases. We obtain classifications and structure results for g almost Abelian, for derived algebra g' of codimension 2 and not J-invariant, for g' totally real, and for g' of dimension at most 2. This leads to a large part of the full classification for two-step solvable SKT algebras of dimension six.
