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 2016-10-06 
Thanks to Marc, I noticed that today's date is a palindrome in two different date formats—DMMYY (61016) and DMMYYYY (6102016).
This made me wonder when there will be another date that is palindromic in multiple date formats, so I wrote a Python script to find out.
Turns out there's not too long to wait: 10 July 2017 will be palindromic in two date formats (MDDYY and MDDYYYY). But before that, there's 1 July 2017, which is palindromic in three date formats (YYMMD, YYMD and MDYY). Most exciting of all, however, is 2 February 2020, which is palindromic in 7 different formats!
The next palindromic dates are shown in the following table. It will update as the dates pass.
 \(n\) Next day with \(\geq n\) palindromic formatsFormats
11 February 2020MMDYY
22 February 2020YYYYMMDD,DDMMYYYY,MMDDYYYY,YYYYMDD,YYMDD,DDMYY,MMDYY
32 February 2020YYYYMMDD,DDMMYYYY,MMDDYYYY,YYYYMDD,YYMDD,DDMYY,MMDYY
42 February 2020YYYYMMDD,DDMMYYYY,MMDDYYYY,YYYYMDD,YYMDD,DDMYY,MMDYY
52 February 2020YYYYMMDD,DDMMYYYY,MMDDYYYY,YYYYMDD,YYMDD,DDMYY,MMDYY
62 February 2020YYYYMMDD,DDMMYYYY,MMDDYYYY,YYYYMDD,YYMDD,DDMYY,MMDYY
72 February 2020YYYYMMDD,DDMMYYYY,MMDDYYYY,YYYYMDD,YYMDD,DDMYY,MMDYY
A full list of future palindromic dates can be found here.

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