Symmetric and strongly symmetric homeomorphisms on the real line with non-symmetric inversion

A quasisymmetric homeomorphism defines an element of the universal Teichmuller space and a symmetric one belongs to its little subspace. We show an example of a symmetric homeomorphism h of the real line $${\mathbb {R}}$$ onto itself such that $$h^{-1}$$ is not symmetric. This implies that the set of all symmetric self-homeomorphisms of $${\mathbb {R}}$$ does not constitute a group under the composition. We also consider the same problem for a strongly symmetric self-homeomorphism of $${\mathbb {R}}$$ which is defined by a certain concept of harmonic analysis. These results reveal the difference of the sets of such self-homeomorphisms of the real line from those of the unit circle.