  Empirical optimal transport (OT) plans and distances provide effective tools
to compare and statistically match probability measures defined on a given
ground space. Fundamental to this are distributional limit laws and we derive a
central limit theorem for the empirical OT distance of circular data. Our limit
results require only mild assumptions in general and include prominent examples
such as the von Mises or wrapped Cauchy family. Most notably, no assumptions
are required when data are sampled from the probability measure to be compared
with, which is in strict contrast to the real line. A bootstrap principle
follows immediately as our proof relies on Hadamard differentiability of the OT
functional. This paves the way for a variety of statistical inference tasks and
is exemplified for asymptotic OT based goodness of fit testing for circular
distributions. We discuss numerical implementation, consistency and investigate
its statistical power. For testing uniformity, it turns out that this approach
performs particularly well for unimodal alternatives and is almost as powerful
as Rayleigh's test, the most powerful invariant test for von Mises
alternatives. For regimes with many modes the circular OT test is less powerful
which is explained by the shape of the corresponding transport plan.
