'non-symmetrical' is formed in English by the negative prefix 'non-' (from Latin 'non', meaning 'not') + 'symmetrical' (from 'symmetry' + '-ical'). 'symmetry' traces to Greek 'symmetria', where 'sym-' (syn-) meant 'together' and 'metron' meant 'measure'.

'symmetria' (Greek) → Latin/Old French forms such as 'symmetria'/'symetrie' → Middle English 'symmetry' → adjective 'symmetrical' (via addition of '-ical'); the prefix 'non-' was later attached in English to form 'non-symmetrical' (also written 'nonsymmetrical').

Initially the root meant 'measured together' (i.e., balanced proportion); 'symmetrical' came to mean 'having balanced proportions or mirror balance,' and 'non-symmetrical' has retained the straightforward negative sense 'not symmetrical.'