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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
122 February 2020MMDDYY
22 March 2020YYMDD,DDMYY
32 January 2021YYMMD,YYMD,MDYY
42 February 2022YYYYMDD,YYMMD,YYMD,DMYY,MDYY
52 February 2022YYYYMDD,YYMMD,YYMD,DMYY,MDYY
61 January 2211YYMMD,DMYYYY,MDYYYY,YYMD,DMYY,MDYY
A full list of future palindromic dates can be found here.

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