Is $\Theta$ symmetric?
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 $x$.