The α-Kakutani substitution rule splits the unit interval into two subintervals of lengths alpha and 1 - α, for a fixed α in (0,1). A simple inflation-substitution procedure produces tilings of the real line and their associated Delone sets. We show that there are precisely five distinct values of min(α, 1 - α) for which these sets are uniformly spread, meaning that they are a bounded displacement of a lattice. The proof of this surprising fact combines the construction and analysis of a related family of primitive substitution tilings, Solomon's criterion for uniform spreadness, and a classification of Pisot-Vijayaraghavan polynomials.