Answer: E

Solution:

The square of the nine-digit repunit is the palindrome $$\begin{aligned} &111{,}111{,}111^2 \\ &= 12{,}345{,}678{,}987{,}654{,}321. \end{aligned}$$

Its digits are $1, 2, \ldots, 9, 8, \ldots, 1,$ so the sum is $$\begin{aligned} &2(1 + 2 + \cdots + 8) + 9 \\ &= 2 \cdot 36 + 9 = 81. \end{aligned}$$

Thus, the correct answer is E.