These unnamed numbers were famous enough to make it to A Mathematician's Apology. After mentioning what you wrote above Hardy writes:

These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals much to a mathematician. The proofs are neither difficult nor interesting—merely a little tiresome. The theorems are not serious; and it is plain that one reason (though perhaps not the most important) is the extreme speciality of both the enunciations and the proofs, which are not capable of any significant generalization.

These numbers are listed in OEIS, and there it is mentioned that

Theorem (David W. Wilson): If reverse(n) = k*n in base 10, then k = 1, 4 or 9.

Off the top of my head I can make the conjecture that none of these numbers contain the digit 3, for example, but I don't know of any interesting mathematical ideas that could be involved in the proof (this doesn't mean anything though, and is of course very subjective).

These unnamed numbers were famous enough to make it to A Mathematician's Apology. After mentioning what you wrote above Hardy writes:

These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals much to a mathematician. The proofs are neither difficult nor interesting—merely a little tiresome. The theorems are not serious; and it is plain that one reason (though perhaps not the most important) is the extreme speciality of both the enunciations and the proofs, which are not capable of any significant generalization.

These numbers are listed in OEIS, and there it is mentioned that

Theorem (David W. Wilson): If reverse(n) = k*n in base 10, then k = 1, 4 or 9.

Off the top of my head I can make the conjecture that none of these numbers contain the digit 3, for example, but I don't know of any interesting mathematical ideas that could be involved in the proof (this doesn't mean anything though, and is of course very subjective).

Edit: To answer your question about places where these numbers have been considered, I will mention this article by Lara Pudwell. The author's views are opposite to Hardy's, and she describes a few related problems and generalizations.