If $\zeta(s)$ is nonzero, but $\zeta(s)\pm\zeta(1-s)=0$, then by the functional equation of the Riemann zeta function we have
$$ \pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\pm \pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)=0.$$
That is, your question is just the Riemann Hypothesis plus a more elementary one similar to your earlier question about the zeros of $\Gamma(s)\pm\Gamma(1-s)$. I would expect that the exact same techniques work here, i.e. one can show by known estimates for the gamma function that all nonreal solutions of the displayed equation lie on $\Re s=1/2$.
EDIT 1. To keep up with new developments I now expect that within the critical strip all nonreal solutions of the displayed equation lie on $\Re s=1/2$. Moreover, it seems reasonable to believe that there are no nonreal solutions with $|\Re s|$ sufficiently large.
EDIT 2. It follows from a generalized Rouché's theorem and Stirling's approximation that there are no nonreal solutions with $|\Re s|$ sufficiently large. More precisely, consider the rectangular contour $C_n$ with vertices $2n\pm it$ and $2n+2\pm it$, where $n>0$ is a large integer and $t>0$ is sufficiently large in terms of $n$. It suffices to show that along $C_n$ we have
$$ \left|\pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)\right|<\left|\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\right|,$$
because this implies that inside $C_n$ there is precisely one solution of the above displayed equation (which must be real by the reflection principle). One can show that the right hand side divided by the left hand side is
$ \gg n^{2n-\frac{1}{2}}(\pi e)^{-2n}$ on the vertical sides of $C_n$, while it is $\gg_n t^{2n-\frac{1}{2}}$ on the horizontal sides of $C_n$. The claim follows.
With the usual definition of a Siegel zero (involving an unspecified constant $C_\varepsilon$ for each $\varepsilon>0$), it is not easy to talk about a "single" Siegel zero unless one decides to fix exactly how $C_\varepsilon$ is to depend on $\varepsilon$.
On the other hand, the classical proof of the prime number theorem also shows that $L(\sigma+it,\chi)$ has no zeroes in the region $\sigma \geq 1-\frac{c}{\log q(|t|+1)}$ for some effective (and very explicit) $c>0$, with at most one exception. This gives an effective prime number theorem in arithmetic progressions
$$ \psi(x; a,q) = \frac{x}{\phi(q)} - \frac{\chi(a)}{\phi(q)} \frac{x^\beta}{\beta} + O( x \exp(-b \sqrt{\log x}))$$
for an absolute and effective constant $b>0$, where $\beta$ is the exceptional zero (if it exists) of the exceptional quadratic character $\chi$. (If there is no exceptional zero, the second term on the right-hand side is simply deleted.) This formula can then be used as a partial but effective substitute for the Siegel-Walfisz theorem for all sorts of number-theoretic applications, e.g. this formula (or something very close to it) is used in all the known effective unconditional proofs of Vinogradov's three primes theorem. In many cases the results are actually easier to prove if the exceptional zero is present. Iwaniec's ICM survey at http://www.icm2006.org/proceedings/Vol_I/16.pdf discusses these issues in more detail.
ADDED LATER: Another interesting phenomenon, first observed by Montgomery and Weinberger, is that the existence of a single Siegel zero $L(\sigma,\chi)=0$ forces many other L-functions $L(s,\psi)$ to have most of their zeroes (at a certain height) arranged on the critical line and to lie close to an arithmetic progression (this type of behaviour is occasionally referred to as the "Alternative Hypothesis", being the extreme opposite to the more commonly believed "GUE hypothesis" but which thus far has proven impossible to completely exclude). Roughly speaking, the reason for this is that if $L(\sigma,\chi)=0$ for some $\sigma$ close to $1$, then the residue of $\zeta(s) L(s,\chi)$ is unexpectedly small at $1$, making the Dirichlet convolution $1*\chi$ much sparser than expected. For any other Dirichlet character $\psi$, $\psi*\chi\psi$ is pointwise dominated by $1*\chi$ and is similarly sparse. This means that the function $L(s,\psi) L(s,\psi\chi)$ is very well behaved for typical $\psi$ and for $s$ near the critical line; indeed, it is dominated by the initial segment of the Dirichlet series $\sum_n \frac{\psi*\psi\chi(n)}{n^s}$ (which is very smooth in $s$), plus the complementary term coming from the functional equation (or equivalently, from Poisson summation), which oscillates at a precise frequency depending on the height of $s$ and the conductor of $\psi$ and $\psi \chi$. The interaction between these two terms is what places the zeroes of $L(s,\psi) L(s,\psi\chi)$, and hence of $L(s,\psi)$, near an arithmetic progression.
Best Answer
A result of Sarnak and Zaharescu, stated in the contrapositive, implies the existence of a complex zero off the critical line for at least one Dirichlet L-function if one has a sufficiently strong Siegel zero: ProjectEuclid link