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We prove that if the squares of two unconditional bases are equivalent up toa permutation, then the bases themselves are permutatively equivalent. Thissettles a twenty year-old question raised by Casazza and Kalton in [Uniquenessof unconditional bases in Banach spaces, Israel J. Math. 103 (1998), 141--175].Solving this problem provides a new paradigm to study the uniqueness ofunconditional basis in the general framework of quasi-Banach spaces. Multipleexamples are given to illustrate how to put in practice this theoreticalscheme. Among the main applications of this principle we obtain the uniquenessof unconditional basis up to permutation of finite sums of quasi-Banach spaceswith this property.

On the permutative equivalence of squares of unconditional bases