  In this paper, we propose optimal tests for circular reflective symmetry
about a fixed median direction. The distributions against which optimality is
achieved are the so-called k-sine-skewed distributions of Umbach and
Jammalamadaka (2009). We first show that sequences of k-sine-skewed models are
locally and asymptotically normal in the vicinity of reflective symmetry.
Following the Le Cam methodology, we then construct optimal (in the maximin
sense) parametric tests for reflective symmetry, which we render
semi-parametric by a studentization argument. These asymptotically
distribution-free tests happen to be uniformly optimal (under any reference
density) and are moreover of a very simple and intuitive form. They furthermore
exhibit nice small sample properties, as we show through a Monte Carlo
simulation study. Our new tests also allow us to re-visit the famous red wood
ants data set of Jander (1957). We further show that one of the proposed
parametric tests can as well serve as a test for uniformity against cardioid
alternatives; this test coincides with the famous circular Rayleigh (1919) test
for uniformity which is thus proved to be (also) optimal against cardioid
alternatives. Moreover, our choice of k-sine-skewed alternatives, which are the
circular analogues of the classical linear skew-symmetric distributions,
permits us a Fisher singularity analysis \`a la Hallin and Ley (2012) with the
result that only the prominent sine-skewed von Mises distribution suffers from
these inferential drawbacks. Finally, we conclude the paper by discussing the
unspecified location case.

Methodology