Does a signal and its autocorrelation ever have the same Fourier spectra?
(a). Yes, a signal and its autocorrelation have the same frequencies. If $X(f)$ is the Fourier transform of the signal and $A(f)$ the Fourier transform of its autocorrelation, then $A(f) = |X(f)|^2$. So, in general, a signal and its autocorrelation don’t have the same Fourier transform, but there is at least one exceptional case:  $x(t) = W \operatorname{sinc}(Wt)$ whose Fourier transform is $X(f) = \operatorname{rect}\left(\frac fW\right)$ and so $A(f) = |X(f)|^2 = X(f)$. Those who wish to generalize this notion to signals whose Fourier spectra are the sum of non-overlapping $\operatorname{rect}$ functions should write out the details for themselves.

All of the above is for finite-energy signals. 

If one is thinking of periodic finite-power signals with Fourier series $x(t) = \sum_{n=-\infty}^\infty c_n \exp(j2\pi nf_0t)$, then the periodic autocorrelation function has Fourier series $\sum_{n=-\infty}^\infty |c_n|^2 \exp(j2\pi nf_0t)$. Fourier series are represented by impulse trains in the frequency domain:
\begin{align}
X(f) &amp;=  \sum_{n=-\infty}^\infty c_n \delta (f-nf_0),\\
A(f) &amp;= \sum_{n=-\infty}^\infty |c_n|^2 \delta (f-nf_0)
\end{align}
and if we wish to continue to use the formula $A(f) = |X(f)|^2$ with the spectra of these periodic signals, then we must assume that
$$c_n \delta(f- nf_0)c_m^* \delta^*(f- mf_0) = \begin{cases}
|c_n|^2 \delta(f- nf_0), &amp; \text{if}~m=n,\\
0, &amp;\text{if}~m\neq n.
\end{cases}$$
Note that we are assuming that $|\delta(f-nf_0)|^2 = \delta(f-nf_0)$ and $\delta(f- nf_0) \delta^*(f- mf_0)=0$ for $m\neq n$, but if we are willing to swallow this bald and unconvincing narrative, then everything is hunky-dory.  Once again, we see that the signal and its autocorrelation have the same frequencies but that their Fourier transforms are generally not equal unless it so happens that $|c_n|^2 = c_n$ for all $n$. Note that this means that for each $n$, $c_n$ must equal $1$ or $0$. Thus, the only real-valued periodic signals whose Fourier transforms equal the Fourier transform of their autocorrelation function are of the form
$a_0 + \sum_{n=1}^\infty a_n 2\cos(2\pi f_0 t),~~ a_i \in \{0,1\}, i = 0, 1, 2, \ldots$