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

<cite authors="Smith, Jason P.">_Smith, Jason P._, [**A formula for the Möbius function of the permutation poset based on a topological decomposition**](http://dx.doi.org/10.1016/j.aam.2017.06.002), Adv. Appl. Math. 91, 98-114 (2017). [ZBL1370.05227](https://zbmath.org/?q=an:1370.05227).</cite>