  Rearrangement-invariance in function spaces can be viewed as a kind of
generalization of 1-symmetry for Schauder bases. We define
subrearrangement-invariance in function spaces as an analogous generalization
of 1-subsymmetry. It is then shown that every rearrangement-invariant function
space is also subrearrangement-invariant. Examples are given to demonstrate
that not every function space on $(0,\infty)$ admits an equivalent
subrearrangement-invariant norm, and that not every subrearrangement-invariant
function space on $(0,\infty)$ admits an equivalent rearrangement-invariant
norm. The latter involves constructing a new family of function spaces inspired
by D.J.H.\ Garling, and we further study them by showing that they are Banach
spaces containing copies of $\ell_p$.
