  Parametric and semiparametric tests of circular reflective symmetry about an
unknown central direction are developed that are locally and asymptotically
optimal in the Le Cam sense against asymmetric $k$-sine-skewed alternatives.
The results from Monte Carlo studies comparing the rejection rates of tests
with those of previously proposed tests lead to recommendations regarding the
use of the various tests with small- to medium-sized samples. Analyses of data
on the directions of cracks in cemented femoral components and the times of gun
crimes in Pittsburgh illustrate the proposed methodology and its bootstrap
extension.
