Indeed, 81 is the first number that requires a long divisor its palindrome is very long. I didn't have the patience to go up to nine digits, as you did. I stopped when the quest for a palindrome reached 8 digits (It took alreadt 32 secs!). In a PC I checked all numbers 1-1000 and, here are the numbers with >=8-digit result (not sulution!): 81, 162, 243, 324, 405, 486, 567, 648, 655, 729, 891, 972. There are no many and may there is a reason for that. For one thing they are multiples of 81, with 1-11 as multipliers ... So, it happens that 81 is the "mystery" number!
Now, it seems there's indeed a palindrome for every number. But how indeed can it be prooven (if it can)?
(My comment has been posted twice - See previous one