A palindromic scale, as I've seen it described on a number of music theory websites, is defined as one whose sequence of intervals is the same when ascending and descending.
For instance: D Dorian (D E F G A B C D) is palindromic because it consists of the interval pattern WHWWWHW, which is a palindrome.
Why should we care? Is there anything that can be done with D dorian that can't be done with C Ionian because it's a palindromic scale?
One possible use is that inversion around a particular pitch axis will create the same scale collection.
Let's use D Dorian, since that was in your example. If we invert D F G A C B E D around tonic (or G♯), the result is still a D-Dorian collection: D B A G E F C D. This is important because if we inverted those same scale degrees in a different collection—let's say D major (D F♯ G A C♯ B E D)—around tonic we'd be left with yet another collection: D Phrygian (D B♭ A G E♭ F C D).
In other words, palindromic collections allow for inversion around tonic while staying within the same collection, which is not true for non-palindromic collections.
