  Webs give a diagrammatic calculus for spaces of $U_q(\mathfrak{sl}_r)$-tensor
invariants, but intrinsic characterizations of web bases are only known in
certain cases. Recently, we introduced hourglass plabic graphs to give the
first such $U_q(\mathfrak{sl}_4)$-web bases. Separately, Fraser introduced a
web basis for Pl\"{u}cker degree two representations of arbitrary
$U_q(\mathfrak{sl}_r)$. Here, we show that Fraser's basis agrees with that
predicted by the hourglass plabic graph framework and give an intrinsic
characterization of the resulting webs. A further compelling feature with many
applications is that our bases exhibit rotation-invariance. Together with the
results of our earlier paper, this implies that hourglass plabic graphs give a
uniform description of all known rotation-invariant $U_q(\mathfrak{sl}_r)$-web
bases. Moreover, this provides a single combinatorial model simultaneously
generalizing the Tamari lattice, the alternating sign matrix lattice, and the
lattice of plane partitions. As a part of our argument, we develop properties
of square faces in arbitrary hourglass plabic graphs, a key step in our program
towards general $U_q(\mathfrak{sl}_r)$-web bases.
