The space complexity of recognising Watson-Crick palindromes
Disclaimer: The following algorithm does not use the standard model for sublinear space complexity (see WP:DSPACE), because it writes to the input tape. Furthermore the set of (Watson-Crick) palindromes is not in $\mathsf{DSPACE}(\mathcal{O}(1)) = \mathsf{REG}$. Nevertheless, it's an in-place solution for many practical purposes (e.g. if each letter is a char in C).

To show that a problem has a specific space complexity, one generally needs to come up with an algorithm that has that space complexity. This may require trial and error, but a better approach is to have a good understanding of the problem you are looking at and a good amount of experience in algorithms and complexity.

For this particular example, there is a third algorithm that does not need any additional space and requires $O(n^2)$ time complexity. This algorithm would be constant space. 

Hint: why use additional space, when you can use the space occupied by the input?

Hint: zip back-and-forth across the string checking one character at a time – use the state of the Turing Machine to remember which character you are checking and erase characters you've already checked.