  We implement the coupled cluster method to very high orders of approximation
to study the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ Heisenberg model on a
cross-striped square lattice. Every nearest-neighbour pair of sites on the
square lattice has an isotropic antiferromagnetic exchange bond of strength
$J_{1}>0$, while the basic square plaquettes in alternate columns have either
both or neither next-nearest-neighbour (diagonal) pairs of sites connected by
an equivalent frustrating bond of strength $J_{2} \equiv \alpha J_{1} > 0$. By
studying the magnetic order parameter (i.e., the average local on-site
magnetization) in the range $0 \leq \alpha \leq 1$ of the frustration parameter
we find that the quasiclassical antiferromagnetic N\'{e}el and (so-called)
double N\'{e}el states form the stable ground-state phases in the respective
regions $\alpha < \alpha_{1a}^{c} = 0.46(1)$ and $\alpha > \alpha_{1b}^{c} =
0.615(5)$. The double N\'{e}el state has N\'{e}el
($\cdots\uparrow\downarrow\uparrow\downarrow\cdots$) ordering along the
(column) direction parallel to the stripes of squares with both or no $J_{2}$
bonds, and spins alternating in a pairwise
($\cdots\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\downarrow\cdots$)
fashion along the perpendicular (row) direction, so that the parallel pairs
occur on squares with both $J_{2}$ bonds present. Further explicit calculations
of both the triplet spin gap and the zero-field uniform transverse magnetic
susceptibility provide compelling evidence that the ground-state phase over all
or most of the intermediate regime $\alpha_{1a}^{c} < \alpha < \alpha_{1b}^{c}$
is a gapped state with no discernible long-range magnetic order.
