$f(x)$ is an analytic function in $\mathbb{R}$ such that $f(-x)f(x)=1$. What else can we find out about $f(x)$?

You have that $f(x)f(-x)=1$ is equivalent on saying that $f(x)= \pm \exp(g(x))$, for some analytic odd function $g$.

In fact, if $g$ is any analytic odd function, then $$e^{g(x)}e^{g(-x)}=e^0=1$$

On the other hand, if $f(x)f(-x)=1$ (WLOG $f(0)=1$), then $f$ never vanishes. So we can think that $f$ is positive everywhere, and taking logarithms $$\log f(x) + \log f(-x) = 0$$ i.e. $\log f$ is an odd function. If we assume $f(0)=-1$, then $\log (-f)$ is odd.