We present a construction of singular rearrangement invariant functionals on Marcinkiewicz function/operator spaces. The functionals constructed differ from all previous examples in the literature in that they fail to be symmetric. In other words, the functional
fails the condition that if
$x\prec \prec \,Y$
(Hardy-Littlewood-Polya submajorization) and
$0\,\le \,\phi \left( x \right)\,\le \,\phi \left( y \right)$
. We apply our results to singular traces on symmetric operator spaces (in particular on symmetrically-normed ideals of compact operators), answering questions raised by Guido and Isola.