If a date is presented in the format dd/mm/yyyy then there are 366 which are palindromes eg 11/02/2011. This is true also if the American date format is used ie mm/dd/yyyy but the palindromes will obviously be different. Here I'm using the English format.
JANUARY palindromes were all in the 11th century: 01/01/1010, 02/01/1020, 03/01/1030 etc up to 31/01/1013.
FEBRUARY's are all in the 21st century: 01/02/2010 through to 29/02/2092.
In MARCH they will be in the 31st century: 01/03/3010 to 31/03/3013.
In APRIL its 41st century: 01/04/4010 to 30/04/4003
MAY 51st century 01/05/5010 to 31/05/5013. JUNE 61st century 01/06/6010 to 30/06/6003.
JULY 71st century 01/07/7010 to 31/07/7013. AUGUST 81st century 01/08/8010 to 31/08/8013.
SEPTEMBER 91st century 01/09/9010 to 30/09/9003. This means that the last palindromic date will be 29/09/9092 assuming (a) that the human race still exists on this date in over 7000 years time and (b) that they are still using a calendar in this format.
OCTOBER palindromes are controversial because the year begins with 01, eg 01/10/0110. 0110 is presumably 110 AD but how were dates expressed at that time ? It seems unlikely that in the 2nd century a date would have been written in this form. However, for the sake of completeness, I'm going to say that if the date format is dd/mm/yyyy then in years prior to 1000, the year number has to start with one or more 0's. So
OCTOBER palindromes run from 01/10/0110 to 31/10/0113 and therefore the earliest palindromic date was 10/10/0101.
NOVEMBER palindromes are all in the 12th century from 01/11/1110 to 30/11/1103 and
DECEMBER palindromes will all be in the 22nd century, 01/12/2110 to 31/12/2113.
FINALLY there are some extra special dates which are double palindromes. Date format dd/mm/yyyy becomes xy/yx/xyyx. When x is the first digit of the date it can only be 0,1,2 or 3. When y is the first digit of the month it can only be 0 or 1. This might suggest that there are eight of these special dates but in fact only six are viable because (1) x and y cannot both be 0 and (2) xy cannot be 31 as there is no month 13. So, six double palindromes: 10/01/1001, 20/02/2002, 30/03/3003, 01/10/0110 (see above), 11/11/1111 and 21/12/2112. The most outstanding of these is clearly 11 Nov 1111. I wonder what happened on this date ?
In England, this date was during the reign of King Henry I but was he, or any of his citizens aware, of its significance ? Maybe not, if they did not use this date format. A date in this form ie same digit repeated eight times, will never happen again as long as dates are written this way.
All this stuff about dates is ultimately meaningless as calendars are completely arbitrary (As indeed are clocks)