  We provide combinatorial tools inspired by work of Warnaar to give
combinatorial interpretations of the sum sides of the Andrews-Gordon and
Bressoud identities. More precisely, we give an explicit weight- and
length-preserving bijection between sets related to integer partitions, which
provides these interpretations. In passing, we discover the $q$-series version
of an identity of Kur\c{s}ung\"oz, similar to the Bressoud identity but with
opposite parity conditions, which we prove combinatorially using the classical
Bressoud identity and our bijection. We also use this bijection to prove
combinatorially many identities, some known and other new, of the
Andrews-Gordon and Bressoud type.
