  In this paper we consider stochastic differential equations with
discontinuous diffusion coefficient of varying sign, for which weak existence
and uniqueness holds but strong uniqueness fails. We introduce the notion of
$\varphi $-strong solution, and we show that under certain conditions on the
diffusion coefficient a $\varphi$-strong solution exists and it is unique. We
also give a construction of a $\varphi$-strong solution and a weak solution in
terms of the notion of \emph{i.i.d. sign choice} introduced in the paper, and
we give an explicit representation of the set of all weak solutions, which in
particular explains the reason for the lack of strong existence and uniqueness.
