Is $\Theta$ symmetric? 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?
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 $x$.