Our calendar system has been widely accepted as a standard despite it's quirks. It is rather systematic and the few anomalies like the leap year too are following a set of rules. As it is, the hard part in the datetime math is usually the time part. And even with that most of the complexity comes from the timezones.
But there are dates that are used in somw contexts which do not follow the usual convention. For example January 0th can be used instead of December 31st in some cases.
Most of such dates are used in some sort of calculations. This way the calculations can be made simpler or they can circumvent certain restrictions in those formulas.
Other uses of non standard dates include the likes of February 30th which is used for "things that will never happen". Another kind of example is May 35th which is use in place of June 4th to get around the censorship (as nothing happened on that date in 1989).