  We prove that if two analytic multicritical circle maps with the same bounded
type rotation number are topologically conjugate by a conjugacy which matches
the critical points of the two maps while preserving the orders of their
criticalities, then the conjugacy necessarily has $C^{1+\alpha}$ regularity,
where $\alpha$ depends only on the bound on the type of the rotation number. We
then extend this rigidity result to $C^3$-smooth bi-cubic circle maps.
