Symmetric multiple zeta values (SMZVs) are elements in the ring of all
multiple zeta values modulo the ideal generated by $\zeta(2)$ introduced by
Kaneko-Zagier as counterparts of finite multiple zeta values. It is known that
symmetric multiple zeta values satisfy double shuffle relations and duality
relations. In this paper, we construct certain lifts of SMZVs which live in the
ring generated by all multiple zeta values and $2\pi i$ as certain iterated
integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty\}$ along a certain closed
path. We call this lifted values as refined symmetric multiple zeta values
(RSMZVs). We show double shuffle relations and duality relations for RSMZVs.
These relations are refinements of the double shuffle relations and the duality
relations of SMZVs. Furthermore we compare RSMZVs to other variants of lifts of
SMZVs. Especially, we prove that RSMZVs coincide with
Bachmann-Takeyama-Tasaka's $\xi$-values.