  We investigate strongly symmetric homeomorphisms of the real line which
appear in harmonic analysis aspects of quasiconformal Teichm\"uller theory. An
element in this class can be characterized by a property that it can be
extended quasiconformally to the upper half-plane so that its complex
dilatation induces a vanishing Carleson measure. However, differently from the
case on the unit circle, strongly symmetric homeomorphisms on the real line are
not preserved under either the composition or the inversion. In this paper, we
present the difference and the relation between these two cases. In particular,
we show that if uniform continuity is assumed for strongly symmetric
homeomorphisms of the real line, then they are preserved by those operations.
We also show that the barycentric extension of uniformly continuous one induces
a vanishing Carleson measure and so do the composition and the inverse of those
quasiconformal homeomorphisms of the upper half-plane.
