See Peter May's "A General Algebraic approach to Steenrod Operations".   Both arise from considering the inclusion of a Sylow $p$-subgroup into $\Sigma_{p^2}$.  There is a universal $\mathbb{Z}$-indexed set of operations with the Steenrod algebra and Dyer-Lashof algebra arising as the quotients acting on the homology of appropriate non-negative (resp., non-positive) complexes.