The question reminds me of the idea behind David W. Jones's 1984 Bangor thesis on "Poly-T-complexes" available here, as follows: we have globular sets, simplicial sets, cubical sets, but what is wrong with pentagons, and why is there a prejudice against rhombic dodecahedra? Also the part of geometric group theory known as "van Kampen diagrams" uses complicated 2-dimensional diagrams to deduce consequences of relations. There are also families of polyhedra such as Stasheff polyhedra.
David defined quite general "cone complexes" with a shelling criterion to avoid some wild examples. In order to model some group theory, and to define polyhedral sets, he also defined "marked cone complexes"; this allows the modelling of the relations $x^3=1$$ x^3=1 $, $xyx=yxy$$ xyx=yxy $, for example.
It turned out recently that the notion of marked complex is equivalent to that of the later defined "discrete vector field" on a complex, for which a web search shows many publications.
See the published version
Jones, D.W. A general theory of polyhedral sets and the corresponding $T$-complexes. Dissertationes Math. (Rozprawy Mat.) 266 (1988) 110.