Problematic challenge of global discernment
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The local sysmmetry operations currently defined originate from those required to create the Archimedean solids and their duals from the Platonic solids. Currently the operations the 13 operations abdegjkmoprst (see table) are defined in terms of the Conway polyhedron notation.. They have been variously extended, notably by Antiprism (Conway Notation transformations / Wythoff-style constructions ).
The following table is a combination of informationm about the operations in Wikipedia (Conway polyhedron notation), Antiprism (Conway Notation transformations / Wythoff-style constructions) and Antitile (Listing of operators and transforms) -- in the light of the table above by Goetschalckx (2020). Much more information is available in each, including polyhedral illustrations, graphs and the characteristic 3x3 matrices. The table below is therefore a simplification to clarify the argument with respect to dialogue. Experimental visualizations of the polyhedra resulting from application of the operations to a cube are presented (as generated by the Antiprism application).
Since that parameters may be added tio the operators. Also, note that, parameters are devised by seeing a sequence pattern based on *one* named operator, and so while the named operator will be in the sequence, there may be more than one sequence that could be reasonably be associated with a parameter for that operator. Operators can be grouped into groups of four (or fewer if some forms are the same) by identifying the operators x, xd (operator of dual), dx (dual of operator), and dxd (conjugate of operator).
The following list of operators has been sorted in ascending order of complexity of transformation, namely to indicate how much greater is the complexity of the polyhedron after the transformation. The table makes provision for indication of the degree to which such a transformation might apply in a dialogue. The terms used to describe the operations are potentially somewhat suggestive of this in metaphorical terms.
The reflect operation changes a left-handed solid to right handed, or vice versa, but has no effect on a reflexible solid.