Number of string permutations that are palindromes
Brachylog (2), 15 bytes

{p.↔}ᶠdl%₆₆₆₀₁₃


Try it online!

Explanation

{p.↔}ᶠdl%₆₆₆₀₁₃
{   }ᶠdl          Count (l) the number of distinct (d) results (ᶠ) obtainable by:
 p                  permuting {the input}
  .                 to produce an output
   ↔                that, if reversed, is still the output
        %₆₆₆₀₁₃   then take that number modulo 666013