  The reversal of a positive integer $A$ is the number obtained by reading $A$
backwards in its decimal representation. A pair $(A,B)$ of positive integers is
said to be palindromic if the reversal of the product $A \times B$ is equal to
the product of the reversals of $A$ and of $B$. A pair $(A,B)$ of positive
integers is said to be polynomial if the product $A \times B$ can be performed
without carry.
  In this paper, we use polynomial pairs in constructing and in studying the
properties of palindromic pairs. It is shown that polynomial pairs are always
palindromic. It is further conjectured that, provided that neither $A$ nor $B$
is itself a palindrome, all palindromic pairs are polynomial. A connection is
made with classical topics in recreational mathematics such as reversal
multiplication, palindromic squares, and repunits.
