arba
I know I am a bit old school here, but as 0000 cannot be allocated to a date unless defined, 2359 is the latest off duty time prior to a day off, as others have defined by CAP371. This being the last chance to comply with the local night definition.
By the old school rules, 0001 would indeed be the start of the day off, however, the duty must finish @ 2359, no later.
In the digital age and that of ACARS, one could say 2359:59 but don't tell the beancounters ;-)
As Daysleeper points out, at 0001 add 23:58 to the rest, or fly 'em again 12 hours later ;-) (sorry Day, couldn't resist)