I think something equivalent to this has been studied in the combinatorics literature.
A CW complex of course has a poset of faces. If the CW complex is regular, then the order complex of the face poset is homeomorphic to the complex. So (assuming regularity, which I haven't checked), your question is equivalent to ordering permutations by subword inclusion up to deletion and monotone reordering. A useful keyword for such studies is permutation patterns.
The lattice of permutations ordered by pattern containment has been studied by Jason Smith. See, for example, the paper
Smith, Jason P., A formula for the Möbius function of the permutation poset based on a topological decomposition, Adv. Appl. Math. 91, 98-114 (2017). ZBL1370.05227.