There is one construction method that leads to the majority of lucky palindromes so far.
Take a prime whose digits are all ones or zeroes, with at most 9 ones, such that its digit reversal is also prime. Their product will not generate any carries, and is a (doubly lucky) palindrome, such as
121121010242121121383121121242010121121 == 11001000011000001011*11010000011000010011
There can also be a two and up to five ones, that just fits (note the central 9):
10201112123150905132121110201 == 100001001010201*102010100100001
And then there are the repunit primes. Combined with 11 they form palindromes of the form 122...221.
12222222222222222221 == 1111111111111111111*11 == 11*1111111111111111111
This happens to be the only lucky palindrome of length 20. I'll wrap up my search and report back with a followup post.