Square of Small-Digit Palindromic Number is Palindromic

Theorem
Let $n$ be an integer such that the sum of the squares of the digits of $n$ in decimal representation is less than $10$.

Let $n$ be palindromic.

Then $n^2$ is also palindromic.

The sequence of such numbers begins:
 * $0, 1, 2, 3, 11, 22, 101, 111, 121, 202, 212, 1001, 1111, \dots$