Let (n, σ, d) denote the variety of all groups defined by the left-normed commutator identity [x1, …, xn] = [x1σ, …, xnσ]d, where σ is a non-identity permutation of {1, …, n}, and d is an integer, possibly negative. It is shown that (n, σ, d) is nilpotent-by-nilpotent if σ ≠ (1, 2), abelian by nilpotent if n > 2, nσ ≠ n, and nilpotent of class at most n + 1 if {1, 2} ≠ {1σ, 2σ}. This improves on a result of E.B. Kikodze that (n, σ, 1) is locally soluble and if {1, 2} ≠ {1σ, 2σ} is locally nilpotent.