Summary: Let \(X\) be a \(d\)-dimensional random vector having zero expectation and unit covariance matrix. T. F. Móri, V. K. Rohatgi and G. J. Székely [Theor. Probab. Appl. 38, No. 3, 547-551 (1993; Zbl 0807.60020)] proposed and studied \(\widetilde{\beta}_{1,d}= \| E(\| X\|^2 X)\|^2\) as a population measure of multivariate skewness. We derive the limit distribution of an affine invariant sample counterpart \(\widetilde{b}_{1,d}\) of \(\widetilde{\beta}_{1,d}\). If the distribution of \(X\) is spherically symmetric, this limit law is \(\lambda\chi_d^2\), where \(\lambda\) depends on \(E\| X\|^4\) and \(E\| X\|^6\). In case of spherical (elliptical) symmetry, we also obtain the asymptotic correlation between \(\widetilde{b}_{1,d}\) and K. V. Mardia’s [Biometrika 57, 519-530 (1970; Zbl 0214.46302)] time-honoured measure of multivariate skewness. If \(\widetilde{\beta}_{1,d}>0\), the limit distribution of \(n^{1/2} (\widetilde{b}_{1,d}- \widetilde{\beta}_{1,d})\) is normal. Our results reveal the deficiencies of a test for multivariate normality based on \(\widetilde{b}_{1,d}\).