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 $mathfrak{g}$ almost Abelian, for derived algebra $mathfrak{g}$ of codimension 2 and not $J$-invariant, for $mathfrak{g}$ totally real, and for $mathfrak{g}$ of dimension at most 2. This leads to a large part of the full classification for two-step solvable SKT algebras of dimension six.
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