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Highlighting the use of critical classes, we consider constituents inKronecker products, in particular of spin characters of the double covers ofthe symmetric and alternating groups. We apply results from the spin case tofind constituents in Kronecker products of characters of the symmetric groups.Via this tool, we make progress on the Saxl conjecture; this claims that for atriangular number $n$, the square of the irreducible character of the symmetricgroup $S_n$ labelled by the staircase contains all irreducible characters of$S_n$ as constituents. We find a large number of constituents in this squarewhich were not detected by other methods. Moreover, the investigation ofKronecker products of spin characters inspires a spin variant of Saxl'sconjecture.

Critical classes, Kronecker products of spin characters, and the Saxl conjecture