For example if $$ f(x)= \Theta (g(x)) $$from the definition of the theta notation, there exist c1 and c2 constants such that$$c_1 g(x) \le f(x) \le c_2 g(x)$$then if only we took the constants $1/c_1$ and $1/c_2$ we could say from the definition that $$ g(x)= \Theta (f(x)) $$Right?
Is $\Theta$ symmetric?
asymptotics;landau notation
Right, except that the constants are actually $1/c_2$ and $1/c_1$.  That is, $$c_1 g(x) \leq f(x) \leq c_2g(x) \Rightarrow \frac{1}{c_2}f(x) \leq g(x) \leq \frac{1}{c_1}f(x)\,.$$Also, remember that the inequalities only apply for large enough $x$.