A chord progression A1 A2 A3,...,An is called symmetric if the relation oftwo succesive transitions Ai A(i+1) is of t he same type. Either harmonic, or melodic or chromatic.
If a symmetric chord progression is such that the relation of the first A1 and last chord An is a chromatic transition, it is called symmetric cycle of chords.
More generally a chord progression partitioned in pairs (A1 B1), (A2, B2),...,(An , Bn) is called pairs-symmetric, if all the transitions (Ai, Bi) are ofthe same type (harmonic, melodic or chromatic) e.g. harmonic and all the transition (Ai, A(i+1)) are of thesame type e.g. chromatic.
The advanage of such symmetric chord progresions is that when composing-improvisiong melody themes on its chord transtitions (e.g. on (A1 A2) or in (A1,B1) for paits symmetric) , by modulating them to the next chord transition we get a melody with symmetric modulations.
But even if the chord progression is not symmetric or pair-symmetric, we may modulate the chord-internal and chord-external melody themes from chord transition to chord transition, with isomorphic ryhthm and ismorphic pitch order structure (topology). Still the melody will sound symmetric. Simply some steps ofthe melody will not always be the same type of intervals but will change from 3rds , to 2nds or 4ths/5ths while the number of notes rhythm and pitch order is preserved.
An example is the song Thalassa platia (wide sea) by M. Hatzidakis where in the first part he is utilizing as symmetric chord progression the chromatic Analuzian cadenza Am->GM->FM->EM7
https://www.youtube.com/watch?v=08a20T78KiA&ab_channel=0204elleni
The refrain of this songs is not a symmetric chord progression, but it could be adopted to have one E.g. melodyc symmetric or chromatic symmetric
E.g. a chromatic symmetric refrain could be Em->Dm->CM->BM7
