No one use the correct convention, which should be year/month/day (the best order for sorting dates by chronological order).
But the European convention, at least, is logical: day/month/year. From the shorter to the lengthier.
The American convention, month/day/year, on the other hand, is illogical, it jumps backward before resuming going on.
It's a bit like numbers in German (128 is ein-hundert-acht-und-zweizig, that is, one-hundred-eight-and-twenty, instead of one-hundert-zweizig-und-acht).
And anyway, 20/04/2004 is not a palyndrome, but a repetition. And in your illogical order, 04/20/2004 is not a palyndrome either. A palyndromic date would be something like 12/02/2021.