Random Walk and 0-1 Law

Question

I am trying to understand the 0-1 Laws and the most common example to use is a random walk. Consider a simple random walk with parameter $p$, $X_n$ that starts at 0 and let us consider the recurrence set $A=\{X_n=0\, i.o\}$. Why is this set not a tail event but symmetric? I know this is the case since I needed the stronger Hewitt 0-1 law rather than the Kolmogorov 0-1 law. On top of that the Hewitt-savage law states that $P(A)=0$ or $P(A)=1$ and provide us with no further distinction. Obviously this will depend on $p$ so I suspect we can classify the events as such, I assume that $p=1/2$ is the deciding point here (as the random sum can oscillate, so $P(A)=1$ for $p=1/2$ and $P(A)=0$ for all other $p$).. But I have no idea how to go about proving it, we can use the events $B=\{ S_n \le -1/2\, a.a\}$ and $C=\{S_n \ge 1/2\, i.o\}$ but I do understand why these events are symmetric and how this helps (even if we apply the 0-1 law, we still dont know what $P(B)$ and $P(C)$ are (apart from it is either 0 or 1).