The approach is very naive. In the case of palindrome, a loop is run to the mid of the string and the first and last characters are matched. If the characters are not similar then the loop breaks and the string is not palindrome otherwise the string is a palindrome.
In the case of symmetry, if the string length is even then the string is broken into two halves and the loop is run, checking the characters of the strings of both the half. If the characters are not similar then the loops break and the string is not symmetrical otherwise the string is symmetrical. If the string length is odd then the string is broken into two halves in such a way that the middle element is left unchecked, and the above same step is repeated.