Time-keeping is based around multiples of six but the calendar isn´t, though there are twelve months. The year is close to being thirty dozen days long. These facts can be used to construct a new calendar.

Starting with the familiar:

Five dozen seconds make a minute.

Five dozen minutes make
an hour.

Two dozen hours make a day.

Now for the slightly less familiar:

Six days make a "week", with two days off a week, numbered zero to five rather than named. The zeroth day of the week is the Sabbath, when people do "zero", i.e. nothing. However, as with the current system these are not part of the date.
Five "weeks" make a "month".

Twelve "months", numbered zero to eleven, starting at the vernal equinox, make one year, corresponding roughlt to the "signs" system of the astrological, but not the astronomical, zodiac. Hence the months also have potential names.
There is a six day "embolic week" at the end of each year which isn´t part of any month. Every four years, this is one day shorter.

A dozen years make a "decade", or a "Cycle of Cathay".

A gross (twelve dozen) years make a "century". The first year of each gross skips a day to make up for the discrepancy arising from the eleven minute discrepancy per year.

Twelve gross of years make a "millenium".

The dating system therefore follows straight on from the time of day, and is in the same order, so it goes YYYY:M:DD:HH:MM:SS - the whole time system is expressed in a single order from large to small. The years are numbered from the first vernal equinox after the beginning of the Julian Date. The first "millenium" is numbered as "zeroth", so we´re now in the third one, beginning with the appropriate digit. With the exception of the embolic week, every date in the year is the same day of the week in all years.

We are now ten gross, eight dozen and one years into the third "millenium", seven months in, and twenty days into that month.

Concerning productivity, the labour-saving effect of having duodecimal number, currency, units of measurement and time is so great that we would be able to afford to have a third of our time off anyway.