Circular Time Delay in signal, effect on phase spectra of DFT
If by phase spectrum you mean the sequence $$\bigr(\angle X[0], \angle X[1], \angle X[2], \ldots, \angle X[N-1]\bigr),$$ then remember that a circular shift
by $M$ in the time domain results in each $X[n]$ being multiplied by $\exp(-j2\pi Mn/N)$, and thus causes a change in $\angle X[n]$; in particular, $\angle X[n]$
decreases by $2\pi Mn/N$.  For the case $M=1$, the multipliers are distinct $N$-th roots of unity, that is, the changes in phase are

$$\Bigr(0, 2\pi\frac{1}{N}, 2\pi\frac{1}{N}, \ldots, 2\pi\frac{N-1}{N}\Bigr)$$

all of which are different numbers.  Thus, each entry in the phase 
spectrum changes (except for the entry for $n=0$). For the case $M=2$ and $N=12$, the multipliers are $\exp(-j2\pi 2n/12) = \exp(-j2\pi n/6)$ and so the changes in phase are

$$\Bigr(0, 2\pi\frac{1}{6}, 2\pi\frac{2}{6}, \ldots, 2\pi\frac{5}{6},
2\pi\frac{6}{6}, 2\pi\frac{7}{6}, \ldots 2\pi\frac{11}{6}\Bigr)$$

But a phase change of $2\pi\frac{6}{6} = 2\pi$ is the same as 
no phase change, a phase change of $2\pi\frac{7}{6}$ is the same
as a phase change of $2\pi\frac{1}{6}$, and so on.  So the displayed
vector above of phase changes is actually 

$$\Bigr(0, 2\pi\frac{1}{6}, 2\pi\frac{2}{6}, \ldots, 2\pi\frac{5}{6},
0, 2\pi\frac{1}{6}, \ldots 2\pi\frac{5}{6}\Bigr)$$

which is two periods of a sequence of period $6$.  Note
each entry in the phase spectrum changes except for $n=0$ and
$n=6$ which do  not change. 

Now suppose that each $X[n]$ is a real number and thus the phase spectrum
is $0$ everywhere. Then, if you circularly shift by one place ($M=1$ in the
time domain, all entries in the phase spectrum will be nonzero
except for the $n=0$ entry.  If instead you circularly shift by
two places ($M=2$) in the time domain, the phase spectrum will have zeroes in
both the $n=0$ and $n=6$ entries.  Note also that $\exp(-j\pi = -1$,
and so for $M=1$, $X[0]$ and $X[6]$ will still be real-valued (though
there will have been a phase change of $\pi$ in $X[6]$) and similarly, for
$M=2$, $X[0], X[3], X[6]$, and $X[9]$ will still remain real-valued
though there will have been a phase change of $\pi$ in $X[3]$ and $X[9]$.