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We exhibit close connections of TCn(X) to the Lusternik-Schnirelmann category of cartesian powers of X, to the cup length of the diagonal embedding X hooked right arrow X-n, and to the ratio between homotopy dimension and connectivity of X. We fully compute the numerical value of TCn for products of spheres, closed 1-connected symplectic manifolds and quaternionic projective spaces. Our study includes two symmetrized versions of TCn(X). The first one, unlike Farber and Grant\u0027s symmetric topological complexity, turns out to be a homotopy invariant of X; the second one is closely tied to the homotopical properties of the configuration space of cardinality-n subsets of X. 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We exhibit close connections of TCn(X) to the Lusternik-Schnirelmann category of cartesian powers of X, to the cup length of the diagonal embedding X hooked right arrow X-n, and to the ratio between homotopy dimension and connectivity of X. We fully compute the numerical value of TCn for products of spheres, closed 1-connected symplectic manifolds and quaternionic projective spaces. Our study includes two symmetrized versions of TCn(X). The first one, unlike Farber and Grant\u0027s symmetric topological complexity, turns out to be a homotopy invariant of X; the second one is closely tied to the homotopical properties of the configuration space of cardinality-n subsets of X. 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