Building functionally complete boolean circuits out of trinary logic
It is not possible to "invert" a value. Here is a list (computed in Python) of all unary functions achievable using your gate (the order of inputs is abc):

aaa
aba
abb
abc
aca
acc
bbb
cbb
cbc
ccc


An inverter would have one of the patterns ba?, c?a, ?cb (depending on which two values you choose to encode True and False), which none of these functions have.

On the other hand, it is possible to compute all monotone functions. Use a for True and c for False. The Or gate is realized as $x \lor y = (xy)$. The And gate is realized as $x \land y = ((((xb)y)b)a)$. (Note that your gate is symmetric but not associative.) There might be other solutions, e.g. using b and c.

Edit: Here is how to extend this to a complete simulation using double wires, according to vzn's suggestion. We represent True by (a,c) and False by (c,a). Negation is accomplished by swapping the two wires. The Or gate is formed by applying the Or gate described above for the first wire, and the And gate for the second wire. The And gate is defined analogously.

Further edit for the curious: here is how to compute all unary functions. We start with the list abc,aaa,bbb,ccc corresponding to the identity function and all constants. We now repeatedly choose two members and compose them, i.e. given functions $f,g$, we compute the function $(fg)$. Eventually the process stabilizes, that is generates no new functions, and we get the ten functions listed above.