Singular symmetric functionals

This is a continuation of the study started in [3]. A linear functional $f$ on a rearrangement invariant space $E$ on $(0, \infty)$ is said to be symmetric if for $x, y\in E$ the condition
$$ \int\limits^t_0x^*(s)sd\le\int\limits^t_0y^*(s)ds,\quad t>0, $$
implies that $f(x)\le f(y)$. A new construction of singular symmetric functionals on the Marcinkiewicz space $M(\psi)$ is presented and studied in detail. A necessary and sufficient condition in terms of $\psi$ is obtained for the seminorms equal to distance to $M(\psi)\cap L_1$ and $M(\psi)\cap L_{\infty}$ to be recoverable in terms of the symmetric singular functionals on $M(\psi)$.