In this paper, we establish six necessary and sufficient conditions for a homeomorphism of \(\mathbb {R}^n\) onto itself to be strongly quasisymmetric. These conditions are quantitative in terms of conformal moduli of disjoint continua as well as the geometric modulus, which was recently introduced by Tukia and Väisälä. Note that all of them are equivalent to bilipschitz continuity with parameters depending also on two fixed points. As an application, we obtain several quantitative characterizations for a homeomorphism of the Riemann sphere \(\overline{\mathbb {R}}^n\) onto itself to be strongly quasimöbius.
