  Two given orbits of a minimal circle homeomorphism $f$ are said to be
geometrically equivalent if there exists a quasisymmetric circle homeomorphism
identifying both orbits and commuting with $f$. By a well-known theorem due to
Herman and Yoccoz, if $f$ is a smooth diffeomorphism with Diophantine rotation
number, then any two orbits are geometrically equivalent. As it follows from
the a-priori bounds of Herman and Swiatek, the same holds if $f$ is a critical
circle map with rotation number of bounded type. By contrast, we prove in the
present paper that if $f$ is a critical circle map whose rotation number
belongs to a certain full Lebesgue measure set in $(0,1)$, then the number of
equivalence classes is uncountable (Theorem A). The proof of this result relies
on the ergodicity of a two-dimensional skew product over the Gauss map. As a
by-product of our techniques, we construct topological conjugacies between
multicritical circle maps which are not quasisymmetric, and we show that this
phenomenon is abundant, both from the topological and measure-theoretical
viewpoints (Theorems B and C).
