TimePickers should conform to ISO8601 5.3.2 on midnight representations, accepting 24:00 as a valid value
ISO 8601 5.3.2 distinguishes between combined date-time representations and time-only (aka time-of-day) representations. The uncombined Time-only type is mapped to the 24-hours of an indeterminate day not on the calendar, which runs from 00:00 to 24:00.
With time-only values,
24:00 is an accepted value in order to distinguish the start of the day from the end of the day.
[A number of databases are IS08601 5.3.2 compliant in regard to time-only types, with separate datatypes for date-time, time, and interval types. PostgreSQL, SQLite, and DB2 among them, and they accept 24:00 as a valid value in the Time datatype columns.]
Timepickers should allow the developer to specify whether incrementing 23:59 by 1 minute goes to 00:00 or to 24:00. It should be possible to set the Timepicker to 24:00 and to get 24:00 from it for submission to an ISO8601-compliant database.
The choice to affix (or not to affix) a time-zone to the time-of-day representation is also a desideratum.