  We revisit the problem of testing for multivariate reflected symmetry about
an unspecified point. Although this testing problem is invariant with respect
to full-rank affine transformations, among the hitherto few proposed tests only
the test studied in [12] respects this property. We identify a measure of
deviation $\Delta$ (say) from symmetry associated with the test statistic $T_n$
(say), and we obtain the limit normal distribution of $T_n$ as $n \to \infty$
under a fixed alternative to symmetry. Since a consistent estimator of the
variance of this limit normal distribution is available, we obtain an
asymptotic confidence interval for $\Delta$. The test, when applied to a
classical data set, strongly rejects the hypothesis of reflected symmetry,
although other tests even do not object against the much stronger hypothesis of
elliptical symmetry.
