*I've combined two answers into one.*

**Bando and Kotschwar's real analyticity in space and time.** As Ian Agol wrote, Shigetoshi Bando, by way of summable derivative of
curvature estimates of Bernstein type, proved that for a solution $(M,g(t)),$
$t\in(0,T)$, to the Ricci flow on a closed manifold, at each $t$ the metric
$g(t)$ is real analytic (in space). These summable estimates and the
consequent real analyticity result can be localized; see arXiv.1111.0355 by
Brett Kotschwar. Also, in arXiv.1210.3083, Kotschwar proves that $g(t)$ is
real analytic in both space and time. This involves careful estimates for
$t^{k+2\ell}|\nabla^{k}\partial_{t}^{\ell}\operatorname{Rm}|^{2}$. As Thomas
Richard wrote, backwards uniqueness was obtained by Kotschwar in arXiv.0906.4920.

**The King-Rosenau solution is the only nonround ancient solution $S^2$.** The following somewhat unrelated facts come to mind regarding long time existence backwards. By this I mean an ancient solution to the Ricci flow on a closed surface. Either it is flat or $R>0$. In the latter case, if it is also Type I, Hamilton proved that the solution must be a round shrinking sphere or its $\mathbb{Z}_2$ quotient. If it is Type II, then Daskalopoulos, Hamilton and Sesum proved that it must be the King-Rosenau solution or its $\mathbb{Z}_2$ quotient. Note that the King-Rosenau solution is rotationally symmetric and invariant under a reflection. Since (loosely speaking) its backwards limits are two opposing cigar steady solitons, one may think of the King-Rosenau solution as a heteroclinic orbit joining two cigars (fixed points modulo conformal diffeomorphisms) to the round sphere (a fixed point modulo scaling).

**The cigar is the only nonflat noncompact 2-dimensional ancient solution.** This additional comment is on $2$-dimensional *noncompact* long time solutions
to the backward Ricci flow, i.e., ancient solutions to the forward Ricci flow.
We indicate a key idea in the proof of Daskalopoulos and Sesum (Intern. Math. Res. Notices 2006) of their result
that any complete noncompact nonflat ancient solution to the Ricci flow
$(M^{2},g(t))$ with bounded curvature and finite width must be a cigar soliton.

Since $R>0$, there exists $f$ such that $\Delta f=-R$ (e.g., see A. Huber,
Comment. Math. Helv. 1957). Because $n=2$ implies $\operatorname{Ric}=\frac
{R}{2}g$, we can write a Bochner formula as
$$
\Delta\left( R+|\nabla f|^{2}\right) =2|\operatorname{Ric}+\nabla^{2}
f|^{2}+4\frac{|\operatorname{div}(\operatorname{Ric}+\nabla^{2}f)|^{2}}{R}+H,
$$
where the trace Harnack $H\doteqdot\Delta R+2|\operatorname{Ric}|^{2}-\frac
{1}{2}\operatorname{Ric}^{-1}(\nabla R,\nabla R)\geq0$ is nonnegative since
the solution is ancient. Remarkably, the right side of the display is the sum
of three nonnegative terms, whereas the left side is a divergence. Now take a
suitable exhaustion $\Omega_{i}$ of $M$. Integrating the display yields that
$\int_{\partial\Omega_{i}}\langle\nabla(R+\left\vert \nabla f\right\vert
^{2}),\nu\rangle ds$ is nonnegative, where $\nu$ is the unit outward normal to
$\partial\Omega_{i}$. It can be shown that this boundary integral tends to
zero as $i\rightarrow\infty$. This implies $\operatorname{Ric}+\nabla^{2}f=0$
and, by the classification of $2$-dimensional steady Ricci solitons, we must
be on a cigar.

See S.-C. Chu (Comm. Anal. Geom. 2007) for the case where the width is not finite.

**Added December 13, 2013**. Consider $2$-dimensional complete noncompact nonflat
ancient solutions with bounded curvature.

(1) By Richard Hamilton, there are no such Type I solutions.

(2) By Sun-Chin Chu, there are no such solutions with infinite width. His work
is based on the works of Wan-Xiong Shi and of Lei Ni and Luen-Fai Tam.

It would be interesting to see if there are other proofs of (2). For example,
assume that $u$ on $M^{2}\times(-\infty,0)$ satisfies $\frac{\partial
u}{\partial t}=-\Delta u$, $u>0$, and $\lim_{t\rightarrow0}u(t)=\delta_{x_{0}
}$.

(i) Can one prove that $\frac{d}{dt}\int uRd\mu\leq0$? Note that $\int
\frac{\partial}{\partial t}(uRd\mu)=\int(-R\Delta u+u\Delta R)d\mu$ is likely
to be zero (can we prove spatial decay of the backward heat kernel?).

(ii) Does there exist $\alpha(t)$ with $\lim_{t\rightarrow-\infty}\alpha(t)=0$
such that $u(x,t)\leq\alpha(t)$? Presumably one needs to use the infinite
width assumption here. Note that $\int(-\frac{\partial}{\partial t}
)(ud\mu)=\int(\Delta u+uR)d\mu$, while $\int uRd\mu\geq0$ is a bad sign for
showing $\int ud\mu$ decays backward in time.

If the answers to (i) and (ii) are yes, then $R(x_{0},0)\leq\alpha(t)\int
Rd\mu(t)\rightarrow0$ as $t\rightarrow-\infty$, which would imply that the
solution is flat, a contradiction.