Summary: Let \(E\) be a symmetric space on \([0,1]\). Let \(\Lambda ({\mathcal R},E)\) be the space of measurable functions \(f\) such that \(fg\in E\) for every almost everywhere convergent series \(g=\sum b_nr_n\in E\), where \((r_n)\) are the Rademacher functions. In [G. P. Curbera, Proc. Edinb. Math. Soc., II. Ser. 40, 119-126 (1997; Zbl 0874.46018)] it was shown that, for a broad class of spaces \(E\), the space \(\Lambda ({\mathcal R},E)\) is not order isomorphic to a symmetric space, and we study the conditions under which such an isomorphism exists. We give conditions on \(E\) for \(\Lambda ({\mathcal R},E)\) to be order isomorphic to \(L_\infty\). This includes some classes of Lorentz and Marcinkiewicz spaces. We also study the conditions under which \(\Lambda ({\mathcal R}, E)\) is order isomorphic to a symmetric space that differs from \(L_\infty\). The answer is positive for the Orlicz spaces \(E=L_{\Phi_q}\) with \(\Phi_q(t)= \exp |t|^q-1\) and \(0<q<2\).