Find the index of reflection of a substring of a palindrome
J, 26 25 bytes
[:(-:@i.&gt;./)[:(*/*#)/.=/~

Try it online!
how
=/~ creates a function table comparing all possible pairs of characters.  Taking 'nasanita' as an example:
1 0 0 0 1 0 0 0
0 1 0 1 0 0 0 1
0 0 1 0 0 0 0 0
0 1 0 1 0 0 0 1
1 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 1 0 1 0 0 0 1

Notice the /-direction diagonals.  If one of those is all ones, it corresponds to an embedded palindrome.
J has an adverb /. which lets us to apply a verb to each diagonal.
We choose the verb (*/*#) -- the sum multiplied by the length.  Thus it will be the length of the diagonal if it's all ones, and 0 otherwise:
1 0 0 0 5 0 0 0 0 0 0 0 0 0 1

i.&gt;./ Find the index of the max element within that list.  In this case the index of 5 is 4.
-:@ And divide it by 2.