'bandboxical' originates from English, specifically formed from the noun 'bandbox' plus the adjectival suffix '-ical', where 'bandbox' meant 'a box for bands (neckbands or collars)' and '-ical' meant 'pertaining to'.

'bandbox' developed in Early Modern English from the combination of 'band' (Old English 'bænd', meaning 'strip' or 'neckband') and 'box' (Old English 'box', meaning 'container'); the adjective 'bandboxical' was later formed by adding the suffix '-ical' to describe things resembling or characteristic of a bandbox.

Initially it referred literally to things relating to a bandbox (a container for bands/collars), but over time it came to describe a person's or object's neat, trim, or fastidiously tidy appearance.