Words that have the same right- and left-associative product
$(\ast)$ If $L$ is a regular language, then $L^R$, the language consisting of the reverse of all words in $L$, is also regular. Take this as an exercise.

How does this help us solve the problem? Let $L_a,L_b,L_c$ be the languages consisting of all strings that evaluate to $a,b,c$ when evaluating from left to right. The language you're interested in is
$$ (L_a \cap L_a^R) \cup (L_b \cap L_b^R) \cup (L_c \cap L_c^R). $$
This shows that if you know how to prove $(\ast)$, then you can construct an NFA for the language in question.

In fact, if you use the idea of the proof of $(\ast)$, then you can probably just go ahead and construct the automaton. So let's consider this. In particular, let's try to construct an NFA for $L_a^R$, the language of all strings that evaluate to $a$ when evaluated from right to left.

The idea is this. Suppose that the first letter you see is $b$. Then the rest of the string must evaluate to $b$ (since $bx = a$ implies $x = b$). Similar reasoning applies when the first letter is $c$. When the first letter is $a$, however, the rest can evaluate to either $a$ or $b$, or be null. With an NFA, we can guess (and later verify our guess).

This hint should give you enough to think about, and hopefully solve the problem.